RAY CLASS GROUPS AND RAY CLASS FIELDS FOR ORDERS OF
NUMBER FIELDS
GENE S. KOPP AND JEFFREY C. LAGARIAS
Abstract. This paper contributes to the theory of orders of number fields. This paper
defines a notion of ray class group associated to an arbitrary order in a number field together with an arbitrary ray class modulus for that order (including Archimedean data),
constructed using invertible fractional ideals of the order. It shows existence of ray classfields corresponding to the class groups. These ray class groups (resp., ray class fields) specialize to classical ray class groups (resp., fields) of a number field in the case of the maximal
order, and they specialize to ring class groups (resp., fields) of orders in the case of trivial
modulus. The paper gives exact sequences for simultaneous change of order and change ofmodulus. As a consequence, we identify the ray class field of an order with a given modulus
as a specific subfield of a ray class field of the maximal order with a larger modulus. We
also uniquely identify each ray class field of an order in terms of the splitting behavior ofprimes.

Contents
1. Introduction
2. Ideals of orders
3. Fractional ideals of orders
4. Change of orders in a number field: extension and contraction of ideals
5. Ray class groups of orders
6. Exact sequences for ray class groups of orders
7. Ray class fields of orders
8. Computations of ray class groups of orders
9. Concluding remarks
Appendix A. Norms of ideals in orders of number fields
References

1
10
17
22
26
31
36
41
44
44
49

1. Introduction
This paper contributes to the theory of orders of number fields. An order O of an algebraic
number field K is a subring of K containing 1 and having finite rank equal to [K : Q] as a
Z-module. Dedekind showed that the ring OK of all algebraic integers in K is the maximal
order, and all other orders O are of finite index in OK .
The object of this paper is to extend to general orders O the notions of ray class groups
and ray class fields for the maximal order OK . A ray class group or field of an order isDate: August 23, 2024.
2020 Mathematics Subject Classification. 11R37 (primary), 11R54 (secondary).
Key words and phrases. class field theory, orders of number fields, ray class fields, ring class fields.
1

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GENE S. KOPP AND JEFFREY C. LAGARIAS

specified by a level datum L = (O; m, Σ) for the field K, in which O is an order of K, m is
an integral ideal of O, and Σ is a set of real places of K. The pair (m, Σ) is called a modulus.
Ray class groups are groups of ray classes, which are sets of fractional ideals satisfying
congruence conditions modulo m and sign conditions at the places in Σ. The set of ray classes
modulo (m, Σ) of a non-maximal order under ideal multiplication forms a monoid (semigroupwith identity) rather than a group, because it includes non-invertible ideal classes. The setof invertible ray classes forms a group, the ray class group of the order.
A ray class field of an order is a certain abelian Galois extension of the number field
K associated to a level datum L = (O; m, Σ). The ray class fields of an order O forma “distinguished” class of abelian extensions of K, cofinal in the partially ordered set ofabelian extensions of K, whose prime decomposition properties over K are described by thecorresponding ray class group and by algebraic properties of the specific order O of K.
1.1. Background. The first complete version of class field theory for number fields was developed by Takagi [56] in 1920, building on work of Weber, Hilbert, Furtwängler, and Fueter;see [28, 30]. In Takagi’s treatment, the ray class fields of a number field K are associatedto ray class groups, which are defined as groups of fractional ideals modulo subgroups ofprincipal ideals satisfying congruence and sign conditions.
The ray class fields of a number field K comprise an infinite set of finite abelian extensionsof K that are cofinal in the set of abelian extensions: That is, every finite abelian extension
is contained in some ray class field. The ray class fields Hm,Σ are attached to moduli (m, Σ),
where m is an ideal of the ring of integers OK , and Σ is a subset of the real embeddings ofK. The extension Hm,Σ /K is ramified at a subset of the primes diving m and the infinite
places in Σ. The ray class field attached to the modulus (OK , ∅) is the Hilbert class field.
Associated to each non-maximal order O of K, there is a separate classical notion of a ringclass field H O , associated to a ring class group Cl(O). When K/Q is an imaginary quadratic
field, the set of ring class fields arise naturally as the fields K(j(τ )) generated by values of
the Klein j-invariant at points τ ∈ K; see Schertz [49, Ch. 6]. The set of all ring class fields
of a field K, obtained by varying the order O, are generally not cofinal in the set of abelian
extensions of K and do not generate the maximal abelian extension K ab . The extension
H O /K is ramified at a subset of the primes dividing the conductor ideal f(O) = (O : OK ).
1.2. Ray class groups for orders. The first object of the paper is the definition of rayclass groups for orders, made in terms of fractional ideals of the order, and the derivationof formulas for computing invariants of such ray class groups, allowing variation of both theorder O and the ray class modulus (m, Σ). Ray class groups for orders specialize to Takagi
ray class groups (when O = OK ) and to ring class groups (when (m, Σ) = (O, ∅)).
In order to define ray class groups, this paper first gives a detailed review of the properties
of integral and fractional ideals of orders, emphasizing their differences from maximal orders,with examples. For integral ideals of non-maximal orders:(1) Ideals do not always factor (uniquely or otherwise) into products of prime ideals.
(2) If an integral ideal a divides b, then b ⊆ a, but the converse need not hold. “Greatest common divisor” and “least common multiple” of ideals are not well defined.
However, the notion of coprimality of two integral ideals is well defined.
(3) Unique factorization of ideals is restored for the set of all integral ideals coprime tothe conductor ideal f(O) of the order, which is the set-theoretically largest ideal of
O that is also an ideal of OK .

RAY CLASSES FOR ORDERS

3

For fractional ideals of non-maximal orders:(1) There exist non-invertible nonzero fractional ideals, so that nonzero fractional ideals
under the ideal product operation form a monoid rather than a group.
(2) The monoid of nonzero fractional ideals J(O) for ideal product may contain nonintegral fractional ideals a having a power ak that is an integral ideal. The ideal ak
may be O, giving non-trivial torsion elements in the group of invertible fractional
ideals J∗ (O). (For the maximal order, J(O) = J∗ (O) is a free abelian group.)(3) There is a notion of coprimality of a fractional ideal and an integral ideal. When a ray
class modulus m contains the conductor ideal f(O), then the set Jm (O) of fractional
ideals coprime to m is a free abelian group.
The paper also treats extension and contraction of integral and fractional ideals between one
order O and a larger order O′ of a fixed number field K. This treatment is intended to be
more broadly useful beyond its application to ray class groups; see Section 1.6 for details.
The ray class group Clm,Σ (O) is a quotient group of the group J∗m (O) of invertible fractional
ideals coprime to the modulus m by a suitable subgroup Pm,Σ (O) of principal ideals. The
modulus m may be any nonzero integral ideal—it is permitted to be non-invertible. For thisreason, the paper necessarily studies the complexities and pitfalls of non-invertible ideals.
More generally, there is a ray class monoid Clmm,Σ (O) under ideal product built from themonoid of fractional ideals Jm (O) coprime to m by modding out by the action of the subgroupPm,Σ (O). Its structure is not considered in this paper and will be treated separately in [33].
The main result of the paper for ray class groups Clm,Σ (O) is an exact sequence relating
ray class groups of orders, in which both the order and the ideal modulus conditions can be
simultaneously varied, given as Theorem 6.5. This exact sequence is obtained using extension
and contraction maps relating integral ideals in two orders O ⊆ O′ . The exact sequence also
yields a formula for the cardinality of a ray class group of an order (Theorem 6.9).
1.3. Ray class fields for orders and class field theory for orders. The second objectof the paper is to establish the existence of ray class fields of orders attached to ray classgroups of orders. These are a distinguished set of abelian extensions of K, whose arithmeticof splitting of primes is described by the given ray class group of the order. We formulate
three results which together comprise a ray class field theory for orders.
The first result states that the ray class field of an order O with modulus (m, Σ) is uniquely
specified by its splitting of primes associated to the principal ray class in the ray class groupwith the given level data. This definition is in the spirit of Weber’s original definition of aclass field in terms of a law of decomposition of prime ideals, motivated by special values ofmodular functions. (See [64, p. 164], [28, p. 266], and [63].)
Theorem 1.1. Let K be a number field, O an order of K, m an ideal of O, and Σ a (possibly
empty) subset of the set of real embeddings of K. Then for the level datum (O; m, Σ), thereO
exists a unique abelian Galois extension Hm,Σ/K with the property that a prime ideal p of
OK that is coprime to the quotient ideal (m : OK ) (as defined in (1.1)) splits completelyOin Hm,Σ
/K if and only if p ∩ O = πO, a principal prime O-ideal having π ∈ O withπ ≡ 1 (mod m) and ρ(π) > 0 for ρ ∈ Σ.
Theorem 1.1 is an existence theorem which, when given the level datum (O; m, Σ), proO
duces an associated ray class field. The map (O; m, Σ) 7→ Hm,Σis neither one-to-one nor
onto: A given abelian extension H/K might be a ray class field for none or many different
triples (O′ ; m′ , Σ′ ).

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GENE S. KOPP AND JEFFREY C. LAGARIAS

To understand Theorem 1.1, it is helpful to compare the quotient ideal (m : OK ) to morefamiliar ideals. Note first that, given two orders O ⊆ O′ , any (integral) O′ -ideal a is automatically an O-ideal. In particular, all OK -ideals are automatically O-ideals for all orders
O of K. Next, recall that, given two O-ideals a, b, the quotient ideal(a : b) := {x ∈ K : xb ⊆ a}.

(1.1)

The quotient ideal (a : b) is an O-ideal, and if in addition b is an O′ -ideal, then (a : b)will also be an O′ -ideal. Thus, given an O-ideal m, the quotient ideal (m : OK ) will be an
OK -ideal, which is the (set-theoretically) largest ideal of OK contained in m.
An important invariant of an order O of an algebraic number field K is its (absolute) conductor f(O) = (O : OK ). The conductor ideal encodes information on all the non-invertible
ideals in the order O; see Lemma 2.13. It is the (set-theoretically) largest integral O-idealthat is also an OK -ideal. It follows that for any integral O-ideal m, we have (m : OK ) ⊆ f(O).
The ideal (m : OK ) satisfies the inclusions (all as OK -ideals(!))f(O)m ⊆ (m : OK ) ⊆ f(O) ∩ mOK ;(1.2)

these inclusions are proven in Lemma 3.9 (taking O′ = OK in its statement). The threeideals in (1.2) have the same prime divisors in the maximal order OK : A prime ideal p of
OK such that p ⊇ f(O)m = f(O)mOK satisfies either p ⊇ f(O) or p ⊇ mOK , and thus,p ⊇ f(O) ∩ mOK .
O/K of an order O for the level datum
The second result locates the ray class field Hm,Σ(O; m, Σ) as falling between two ray class fields on the maximal order.
Theorem 1.2. For an order O in a number field K and any level datum (O; m, Σ), there
OKOKO⊆ H(m:O.
are inclusions of ray class fields HmO⊆ Hm,Σ
K ,Σ
K ),Σ
O
In the special case m = O, we have (m : OK ) = f(O). The smallest ray class field HO,∅ofthe order O is the ring class field associated to O, which always contains the (wide) Hilbertclass field of OK and whose ramification over K occurs only at prime OK -ideals containing
the conductor ideal f(O).
It follows from Theorem 1.2 and (1.2), together with standard class field theory, that theset of all ray class fields of a fixed order O is cofinal in the set of all finite abelian extensionsof K. Theorem 1.2 is obtained as a special case of a general result relating ray class fieldsof two given orders O ⊆ O′ , given as Theorem 7.5.
The third result adapts Artin reciprocity to our setting to give a correspondence between
class groups and Galois groups of class fields. The correspondence asserts that the ray classO
field Hm,Σof an order O is associated to an appropriate ray class group Clm,Σ (O) in such aO
way that a Galois correspondence holds: Gal(Hm,Σ/K) ∼= Clm,Σ (O) as abelian groups.

Theorem 1.3. For an order O in a number field K and any level datum (O; m, Σ), with asOsociated ray class field H0 := Hm,Σ, there is an isomorphism ArtO : Clm,Σ (O) → Gal(H0 /K),
uniquely determined by its behavior on prime ideals p of O that are coprime to f(O) ∩ m,having the property that
ArtO ([p])(α) ≡ αq (mod P) ,where P is any prime of OH0 lying over pOK , and q = pj is the number of elements in thefinite field O/p. For any (not necessarily prime) ideal a of O coprime to f(O) ∩ m,
ArtO ([a]) = Art([aOK ])|H0 ,(1.3)

RAY CLASSES FOR ORDERS

5

where Art : Cl(m:OK ),Σ (OK ) → Gal(H1 /K) is the usual Artin map in class field theory, withOK
H1 = H(m:O
being a (Takagi) ray class field for the maximal order OK , and H0 ⊆ H1 .
K ),ΣIn Theorem 1.3, the set of prime ideals coprime to f(O) ∩ m includes all but finitely manyof the prime ideals of O.
For the classical case of the maximal order OK , the Artin map Art(·) on the right side of(1.3) is given in Neukirch [46, Ch. VI, Thm. 7.1] for any ray ideal m, in the case of Σ = Σmax ,the set of all real places of K. Additionally, the special case m = OK with Σ = Σmax produces
the narrow Hilbert class field (termed by Neukirch the big Hilbert class field) of K, denoted
HOOKK,Σmax ; compare [46, Ch. VI, Prop. 6.8]. The alternative choice of1 the minimal set Σ = ∅of real places produces the original (or wide) Hilbert class field (called by Neukirch the small
Hilbert class field) of K, denoted HOOKK,∅ . The wide class field has Galois group isomorphic
to the ideal class group of OK ; see [46, Ch. VI, Prop. 6.9]. The difference between wide andnarrow Hilbert class fields exhibits the role of the real places Σ in the level datum.
Our three main theorems do not provide an independent development of global classfield theory; their proofs are completed using the existing global class field theory. Their
contribution is to identify a new set of distinguished abelian extensions of K whose structure
encodes properties of the arithmetic of the ray class groups attached to a fixed order O of
K (rather than the maximal order).
The proofs of these these results proceed on the ray class group side. We identify the
Clm,Σ (O) with a certain quotient group of a Takagi ray class group of the maximal order.
The exact sequences in Theorem 6.5 are used. Our definition (Definition 7.1) of the rayclass field associated to the level datum (O; m, Σ) is formulated using this identification.
Theorem 1.1, Theorem 1.2 and Theorem 1.3 are then obtained via the main correspondence
and existence theorems of global class field theory in a form given in Section 7.
1.4. Applications. The constructions in this paper are motivated by several mathematicalphenomena involving ray class groups and ray class fields of orders that arise from outsideof pure class field theory. We briefly describe three of them.
(1) Special configurations of complex lines, called SIC-POVMs (symmetric informationally complete positive operator-valued measures) or SICs, are of interest in quantum
information theory and also suggest a geometric approach to explicit class field theory
for real quadratic fields that fundamentally involves non-maximal orders. A SIC is a
generalized quantum measurement (a POVM ) with restrictive information-theoretic
properties that make it optimal for certain protocols in quantum information processing. It is equivalent to a maximal set of d2 complex lines in Cd that are “equiangular”
with respect to the Hermitian inner product. SICs are known to exist in at least dimensions d ≤ 53 and are conjectured by Zauner [66, 67] to exist in all dimensions.
Recently, a surprising connection has been made between SICs and the explicit
class field theory of real quadratic fields [2, 3, 31]. The connection was originallydiscovered through numerical experimentation. Work by the current authors andothers [1, 34] uses ray class groups and ray class fields of real quadratic orders in
a conjectural framework for classifying SICs. That work also indicates SICs andgeneralized SICs may be used to study abelian extensions of real quadratic fields.
1Neukirch does not introduce the real place parameters Σ, and his definition [46, Ch. VI, Defn. 6.2] of rayclass groups corresponds to the choice Σ = Σmax .

6GENE S. KOPP AND JEFFREY C. LAGARIAS(2) Class field theory for imaginary quadratic orders arises naturally in the theory ofcomplex multiplication (CM) for elliptic curves. In particular, if E is an elliptic curveOwith CM by O having a Weierstrass equation with coefficients in Q(j(E)) = HO,∅,O
then the n-torsion points of E have coordinates in the field HnO,∅ ; see [8, Thm. 1.4].
A related construction describes elliptic units in abelian extensions of imaginary
quadratic fields as special values of modular functions called modular units [35]; thisconstruction also naturally extends to non-maximal orders.
Ray class fields of orders of higher degree arise in the study of higher-dimensionalabelian varieties with CM. Pathologies of non-maximal orders of degree greater than2, such as potentially being non-Gorenstein, create obstructions to proving theorems
in CM theory [11].
(3) Ray class groups and ray class fields of orders appear naturally in approaches toHilbert’s 12th Problem for real quadratic and complex cubic fields. For example,
when replacing the complex upper half plane by the p-adic Drinfeld upper half plane,
as in the work of Darmon and Dasgupta on elliptic units for real quadratic fields [19]and the work of Darmon, Pozzi, and Vonk [20, 21] on rigid mermomorphic cocycles,
non-maximal orders arise in the study of “singular moduli” in the same way as they
do in the classical theory of complex multiplication. Non-maximal orders are needed
to describe arbitrary real multiplication (RM) values of rigid meromophic cocycles.
Complex meromorphic modular cocycles introduced by the first author [32] have
RM values that are (essentially) the classical Archimedean Stark class invariants
conjectured to be algebraic units by Stark in the real quadratic case. The RM
values studied are naturally parametrized by elements of ray class groups (moregenerally, ray class monoids) of orders, and they are conjectured to lie in ray classfields of orders. We also expect the elliptic gamma functions connected by Bergeron,
Charollois, and Garcı́a to Stark units over complex cubic fields [7] to have special
values parameterized by ray classes of general complex cubic orders, which generate
ray class fields of those orders.
The treatment of integral and fractional ideals of orders, and extension and contraction
maps, given in Sections 2 to 4, is intended to apply more broadly, outside its use in defining
ray class groups of orders. The related concept of ideal lattices is foundational to various
versions of lattice based-cryptography [38, 44, 45] and ring-based schemes proposed for homomorphic encryption [26, 39]. Ideal lattices are identified with ideals in polynomial rings
Z[x]/(f (x)), which for irreducible monic f (x) correspond to ideals of the monogenic orderO = Z[θ] generated by a root of f . An ideal lattice can be encoded as an integer matrix associated to an integral ideal of an order, as studied by Taussky [58–60]; see also Taussky’s earlier
work [57]. A number-theoretic perspective on ideal lattices is given by Bayer-Fluckiger [5,6].
1.5. Prior work. There has been an extensive algebraic study of the structure of orders of
number fields, beginning with Dedekind [22] in 1877; see also [23]. Two general references
are Stevenhagen [54] and Neukirch [46, Ch. 1, Sec. 12]. Neukirch views orders of number
fields as number rings with “singularities” at the primes dividing the conductor ideal, by
analogy with geometric interpretations of subrings of function fields (e.g., F [t2 , t3 ] ⊆ F (t) as
the coordinate ring of the cuspidal cubic).
An important 1962 paper of Dade, Taussky, and Zassenhaus [18] presented fundamental
results on the structure of invertible fractional ideals, class groups of orders, and the classRAY CLASSES FOR ORDERS

7

monoids obtained when including non-invertible ideals. They developed a structure theory
for one-dimensional Noetherian domains [18, p. 32]. (An integral domain D has dimension
one if and only if all nonzero prime ideals are maximal. Orders in number fields form a strict
subclass of one-dimensional Noetherian integral domains.) Dade, Taussky, and Zassenhaus
gave a general definition of fractional ideals valid for all integral domains D (with quotientfield denoted K) in [18, Defn. 1.1.6]. The set J(D) of all such fractional ideals of D isclosed under four operations: +, ·, ∩, ( : ), in which · is ideal multiplication and ( : ) is theideal quotient as defined by (1.1); see [18, Prop. 1.10]. They note the set J(D) carries the
structure of a semigroup under ideal multiplication. Dade, Taussky, and Zassenhaus definea D-order to be any fractional ideal a of D that is also an integral domain [18, p. 32]. Everyfractional ideal a has an associated D-order ord(a) := (a : a), which in other contexts iscalled its multiplier ring. For Noetherian integral domains, they define invertible fractional
ideals and characterize them using ideal quotient [18, Defn. (p. 41), Prop. 1.3.6].
Class field theory has gone through many versions. We use the version using ray classes,
formulated in the work of Takagi, Hasse, and Artin. A useful treatment is given in Cohn [15].
An appendix of Cohn’s book gives Emil Artin’s 1932 lectures.
Computational class field theory, generally using ray classes, is especially important inapplications (such as those mentioned in Section 1.4). Cohen and Stevenhagen have written
a useful survey [14].
Ring class groups and ring class fields go back to fundamental work of Weber in 1897–
1898 [63], motivated (in part) by complex multiplication; see his books [61, 62, 64]. The
ring class groups associated to orders of imaginary quadratic fields appear in the theoryof complex multiplication, because ideal classes of those orders are classified by homothety
classes of lattices in C having a given endomorphism ring O ̸= Z; see [17,√ Cor. 10.20].
The Weber prime splitting criteria for ring class fields of orders O = Z[ −n] (n > 0)
over imaginary quadratic fields are expressible in terms primes represented by the principal
quadratic forms x2 + ny 2 . Explaining this connection is the main objective of the book of
Cox [17], as stated in [17, Thm. 9.2].
It is known that the compositum of all ring class fields of a field K need not be equal
to the maximal abelian extension K ab . That is, there can be abelian extensions of K thatare not√ contained in any ring class field. In 1914, Fueter [25, p. 178] showed that the field
Q(i, 4 1 + 2i) is not contained in any ring class field of Q(i); see also Schappacher [48, p.
258]. Moreover, Bruckner [9, Satz 8] showed for a quadratic field K that the compositumof ring class fields for K is the compositum of all Galois fields L containing K for which
Gal(L/Q) is a generalized dihedral group. The case of imaginary quadratic fields is also
treated in Cox [17, Thm. 9.18, Cor. 11.35].
In 2015, Lv and Deng [37] treated ring class fields for arbitrary orders in number fields ina classical setting, corresponding to the “unramified” case (O; m, Σ) = (O; O, ∅). (The term
“unramified” here refers to the fact that the modulus is trivial; the ring class field usuallyis a ramified extension at the primes containing the conductor.) An extension to general
number rings (allowing inversion of arbitrary sets of nonzero elements) was given in 2018 by
Yi and Lv [65].
There was significant further work on of class field theory for orders in the general “ramified” case, in situations related to complex multiplication. In 1935, Söhngen [50] constructed a “class field theory for orders” for imaginary quadratic fields, with explicit generators given by special values of Weber functions, which is described in detail in the book

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GENE S. KOPP AND JEFFREY C. LAGARIAS

of Schertz [49, Ch. 3, Ch. 6.2]. The ray class field theory for imaginary quadratic orders
has been applied to CM theory for elliptic curves; we mention Bourdon and Clark [8] andLozano-Robledo [36, Sec. 3].
In another direction of generalization, in the late 1980’s, Stevenhagen [51, 52] formulated
an abstract development of unramified class field theory for orders, which extends beyondorders of number fields; see also [51]. In the case of orders of number fields, his results would
specialize to the “unramified case” (O; m, Σ) = (O; O, ∅) treated here. In 2001 Stevenhagen[53, Sec. 4] recast much of Söhngen’s class field theory for imaginary quadratic orders into a
profinite, idèlic framework.
Very recently Campagna and Pengo [10, Sec. 4] developed a class field theory for ordersof general number fields using an idèlic framework. They define ray class fields of orders by
specifying a particular idèlic unit group attached to a level datum (O; m, ∅), which defines
the associated class field by the idelic main theorem of class field theory. They did notconsider ray conditions at the Archimedean places, since their interest was CM fields. The
PhD thesis of Pengo [47, Sec. 6] formulated an idèlic definition of ray class fields for orders
in the general case.
1.6. Contents of the paper. This paper works in the classical framework of integral andfractional ideals. Schertz [49, p. 82] used the classical framework in treating the theory ofcomplex multiplication “because the classical language marries well with singular values of
elliptic and modular functions that are essentially dependent on ideals.” These functions
are used in explicit class field theory. The classical ideal viewpoint is appropriate also forformulating the structures of monoids of all ray classes, allowing non-invertible classes, whicharise in applications [32, 33].
Section 2 gives an extensive treatment of integral ideals of an order, including invertible
and non-invertible ideals, needed for the constructions of ray class groups and maps between
them. It emphasizes the special features of non-maximal orders and contains many examples.
This section is needed because many calculations and proofs in the paper use possibly noninvertible integral ideals, sometimes in semilocal rings obtained by inverting elements coprime
to a single auxiliary ideal d. Each non-invertible integral ideal contains some non-invertible
prime ideal, and the non-invertible prime ideals are exactly those prime ideals containing the
conductor ideal of the order. The set of integral ideals of an order of a number field coprime
to a modulus ideal m forms a monoid under ideal multiplication.
Section 3 treats fractional ideals of an order and gives further examples of behavior special
to non-maximal orders. The ideal quotient of two fractional ideals is always well-defined;
however, not all fractional ideals are invertible. There exist invertible ideals that are torsion
elements of the invertible ideal group. The monoid of all nonzero fractional ideals coprime
to the conductor ideal of the order is a free abelian group.
Section 4 studies, inside a fixed number field K, the effect of change of order on the
structure of ideals, via the contraction and extension maps on integral and fractional ideals.
A detailed treatment is needed for showing good behavior away from the conductor idealand for proving exact sequences of ray class groups in Section 6. A ring-theoretic subtlety isthat the contraction map con(a) = a ∩ O is not a monoid homomorphism for integral ideals.
We show the contraction map when restricted to integral ideals coprime to the (relative)
conductor ideal fO′ (O) is a monoid homomorphism. Using this property, we are able todefine a contraction map on fractional ideals coprime to f(O), which is a homomorphism but
no longer agrees with the formula con(a) = a ∩ O.

RAY CLASSES FOR ORDERS

9

Section 5 defines ray class groups of orders for a level datum (O; m, Σ). It relates suchgroups under extension and contraction of order. Lemma 5.12 is a key technical result,showing that ray class groups do not change when adding coprimality conditions to an auxiliary modulus d, permitting comparison of different orders and different moduli. Section 5.4
determines a surjective map from a general ray class group of an order to a particular ray
class group of the maximal order.
Section 6 gives exact sequences relating ray class groups under change of order and changeof modulus. It first analyzes the effect of change of order O ⊆ O′ on unit groups and principal
ideal groups, in Proposition 6.4. The exact sequence of Theorem 6.5 in Section 6.2, which
relates unit groups and class groups of different orders and moduli, is the main formula ofthis paper for applications. Section 6.3 gives a formula for a generalized class number, thatis, the cardinality of a given ray class group of an order. This formula generalizes a formulain Neukirch [46, Thm. I. 12.12] in the ring class group case.
Section 7 gives the construction of ray class fields of orders. For this purpose, it is necessary
to obtain a given ray class group of an order O as a quotient group of a particular Takagi
ray class group of the maximal order OK . This is done in Section 7.1. The desired rayclass field of the order is then identified as a subfield L of a Takagi ray class field H1 . It is
Art(ker(ψ))
defined as the fixed field L = H1of the kernel of the map ψ in (7.1) acting via theArtin map. Section 7.2 recalls the main existence theorems of class field theory in a suitableform encoding both the formulation of Takagi (in terms of prime splitting) and of Artin (interms of an isomorphism between ray class group and Galois groups). Section 7.3 proves
Theorem 1.1 by identifying the map ψ in (7.1) with the contraction map between fractional
ideals of the maximal order OK and the given order O. Section 7.4 states and proves a
result generalizing Theorem 1.2, replacing the maximal order OK with a general order O′
with O ⊆ O′ ⊆ OK . Section 7.5 proves Theorem 1.3.
Section 8 presents examples of ray class groups and ray class fields of quadratic orders.
Section 9 presents some remarks on extending results of the paper.
Appendix A discusses norms of ideals of an order and the (consistent) extension of thenorm to fractional ideals of an order. The norm is multiplicative when multiplying two
fractional ideals, at least one of which is invertible, but is not multiplicative in general.
1.7. Notation.
• O = arbitrary order of a number field K.
• f(O) = (O : O) = conductor ideal of O.
• O′ = another arbitrary order of K satisfying O ⊆ O′ .
• fO′ (O) = (O : O′ ) = relative conductor ideal.
• ord(a) = (a : a) = multiplier ring of the O-ideal a, which is an order O′ .
• I(O) = monoid of integral ideals of the order O.
• I∗ (O) = submonoid of I(O) of integral ideals invertible as fractional ideals of O.
• m = general integral O-ideal; finite part of the ray class modulus.
• Σ = subset of the set of real embeddings of K; infinite part of the ray class modulus.
• Im (O) = monoid of integral ideals of O coprime to the integral ideal m.
• I∗m (O) = submonoid of Im (O) of integral ideals invertible as fractional ideals of O.
• J(O) = monoid of all fractional ideals of O
• J∗ (O) = group of invertible fractional ideals of O.
• Jm (O) = monoid of fractional ideals coprime to the (nonzero) integral ideal m of O.

10

GENE S. KOPP AND JEFFREY C. LAGARIAS• J∗m (O) = group of invertible fractional ideals coprime to m.
• P(O) = group of nonzero principal fractional ideals αO, with α ∈ K × .
• Pm,Σ (O) = group of nonzero principal fractional ideals αO, with α ∈ K × , α ≡1 (mod m), and ρ(α) > 0 for ρ ∈ Σ.
• Pdm,Σ (O) = subgroup of Pm,Σ (O) of all αO = (a : b) = ab−1 an ideal quotient of
invertible integral ideals a, b, each coprime to a given integral ideal d of O.
• Clm,Σ (O) = ray class group of order O with modulus (m, Σ).
• L := (O; m, Σ) abbreviates a (ray class) level datum, where O is an order of a number
field K, m is an integral O-ideal, and Σ is a subset of the real places of K.
2. Ideals of orders
Let OK be the maximal order of all algebraic integers in a number field K. Then OK
is a Dedekind domain, having unique prime factorization of nonzero integral ideals. All
nonzero fractional ideals are invertible, and they form a free abelian group. One has an idealclass group defined as the quotient of the group of nonzero fractional ideals by the groupof nonzero principal ideals. One can define ray class groups by restricting to ideals coprime
to a modulus m and quotienting by the group of principal prime ideals having a generator
α ≡ 1 (mod m) and with some positivity conditions at a subset Σ of real places.
Non-maximal orders of a number field are never Dedekind domains; rather, they are
one-dimensional Noetherian integral domains that are not regular rings. The ideal theoryof non-maximal orders has notable differences from that of the maximal order. Not all
integral ideals factor into prime ideals (uniquely or otherwise). There exist non-invertible
integral ideals. “Greatest common divisor” and “least common multiple” of ideals are notwell-defined, although there is a notion of coprimality of two ideals. The failure of uniquefactorization into prime ideals is restored on restricting to ideals coprime to the conductor
ideal f(O) = (O : OK ). In addition, the monoid of invertible integral ideals need not be free.
In the rest of Section 2 and in Section 3, we state and prove required foundational results atvarying levels of generality, always restricted to commutative rings with unity. In decreasing
generality, these include integral domains, Noetherian integral domains, Noetherian integral
domains of dimension one, and orders O of algebraic number fields.
Compared to more general integral domains, orders of a number field K have strong
finiteness properties arising from the Q-lattice structure on K. A full rank Z-lattice of K is
any Z-module Λ = α1 Z + α2 Z + · · · + αn Z with each αi ∈ K, having Z-rank n = [K : Q].
The multiplier ring ord(Λ) of a full rank Z-lattice Λ of a number field K is given byord(Λ) = (Λ : Λ) := {α ∈ K : αΛ ⊆ Λ},and it is an order of K. The multiplier ring is the largest order (as a set) such that Λ isa fractional ideal of that order. Each order O of K is the multiplier ring of some full rank
Z-lattice, namely, itself: O = ord(O). Finiteness properties of orders of number fields include
the well-known finiteness of the class group Cl(O) of an order.
2.1. Integral ideals, prime ideals, and primary ideals. We start with a general integral
domain D. An integral ideal (or simply an ideal or D-ideal ) a of D is an D-submodule a ⊆ D.
The D-ideal product ab of two D-ideals a, b is the D-ideal)(X
ab = a · b =αj βj : αj ∈ a, βj ∈ b .
(2.1)
jRAY CLASSES FOR ORDERS

11

We let I(D) denote the set of integral ideals of an integral domain, which forms a monoid
for the operation of D-ideal product, with D as the identity. A D-ideal p is prime if xy ∈ p
implies x ∈ p or y ∈ p, and p ̸= D. Additionally, there is a notion of coprimality of D-ideals.
Definition 2.1 (Coprimality of integral ideals). An integral ideal a ⊆ D of an integral
domain D is said to be coprime (or relatively prime) to another integral ideal m ⊆ D if
a + m = D.
If a, b are both coprime to m, then their product ab is coprime to m, because
D = (a + m)(b + m) = ab + am + bm + mm ⊆ ab + m ⊆ D.
We now specialize D to be a Noetherian integral domain. Thus, all ideals of D are finitely
generated.
Commutative Noetherian rings generally do not have unique factorization into productsof powers of prime ideals; they possess a weaker form of decomposition of ideals underintersection, called primary decomposition [4, Ch. 4]. A primary ideal q is an ideal suchthat, if xy ∈ q, then either x ∈ q or y n ∈ q for some n ≥ 1, and q ̸= D. Any power ofa prime ideal is primary. (Indeed, any power of a primary ideal is primary.) The primary
decomposition for a Noetherian ring is a decomposition of an ideal into an intersection ofprimary ideals. In a commutative Noetherian ring, all ideals have a primary decomposition(Lasker–Noether theorem); however, a primary decomposition is not necessarily unique.
For one-dimensional Noetherian domains, stronger results hold. One-dimensional Noetherian domains are Noetherian domains for which all nonzero prime ideals are maximal.
The class of one-dimensional Noetherian domains includes all orders of number fields and isclosed under localization and under completion. For a one-dimensional Noetherian domain
D, rimary decompositions of ideals exist and are unique. In addition, the primary decomposition given as an intersection of ideals coincides with its decomposition as a product of
the same primary ideals. To state the result precisely, recall that the radical of an ideal israd(m) := {x ∈ A : xn ∈ m for some n ≥ 1}.
The radical rad(q) of a primary ideal is the unique prime ideal p containing q. We say thatsuch a primary ideal q is associated to the prime ideal p, or alternatively, that q is p-primary.
Proposition 2.2 (Primary decomposition in dimension 1). Let D be a commutative Noetherian integral domain in which all nonzero prime ideals are maximal (i.e., D has Krull
dimension 1). Then
(1) Every non-zero ideal m in D has a unique primary decomposition
\
m=
qi ,
i

in which qi are primary ideals whose radicals pi = rad(qi ) are pairwise distinct.
(2) The primary decomposition agrees with its product decomposition
Y
m=
qi .
(2.2)
i

Proof. This is [4, Prop. 9.1]. The last assertion (2.2) is established in its proof.

□

Proposition 2.2(2) gives a form of unique factorization into pairwise coprime ideals, inwhich every factor is primary.

12

GENE S. KOPP AND JEFFREY C. LAGARIAS

A prime ideal p is called regular if all the p-primary ideals are powers of p; it is singular otherwise. The singular prime ideals of orders of number fields are characterized in
Lemma 2.13. Each order O of K has finitely many singular prime ideals; the maximal order
OK is the only order having no singular prime ideals.
2.2. Invertible integral ideals of orders of number fields. Recall that there is associated to each integral O-ideal a of a number field a multiplier ringord(a) := (a : a) = {x ∈ K : xa ⊆ a}.
Necessarily O ⊆ ord(a) ⊆ OK . All orders O′ between O and OK occur this way; one may
choose γ ∈ K × so that a = γO′ ⊆ O, and a is then an integral O-ideal having ord(a) = O′ .
Definition 2.3 (Invertible integral ideal). An integral ideal a of O is invertible if there exists
another integral O-ideal b and a nonzero γ ∈ O such that the O-ideal product ab = γO.
Otherwise a is non-invertible.
The invertibility property is preserved under O-ideal product. If ac = λO and bd = µO,
then (ac)(bd) = λµO, so ab is invertible. This statement also has a converse.
Lemma 2.4. Let O be an order of a number field.
(1) If c is an invertible integral O-ideal, and c = ab as O-ideals, then both a and b are
invertible O-ideals.
(2) If a, b, c are O-ideals (invertible or not) and ord(c) = O, then c = ab implies ord(a) =ord(b) = O. In particular, all invertible integral O-ideals a have ord(a) = O.
Proof of (1). Given c = ab is invertible, then there is an integral ideal d with cd = γO. Now
a is invertible since a(bd) = cd = γO, and similarly for b.
□
Proof of (2). Given c = ab, we have ord(a) ⊆ ord(ab) = ord(c). Since O ⊆ ord(a) ⊆ ord(c)and ord(c) = O by hypothesis, we have ord(a) = O; similarly ord(b) = O. For invertible
ideals we have ac = γO. Since ord(γO) = O, we deduce ord(a) = O.
□
We let I∗ (O) denote the monoid of invertible integral ideals under ideal multiplication.
All nonzero principal ideals a = αO ∈ I∗ (O), because (αO)(O) = αO.
Not all ideals of a general order O are invertible; a necessary condition for invertibility ofan O-ideal a is that ord(a) = O, as given in Lemma 2.4(2). The next example shows thisnecessary condition is in general not sufficient.
Example 2.5. [Non-invertible ideal q of O with ord(q) = O] (This phenomenon occurs only
for number
fields K with [K : Q] ≥ 3 having a non-Gorenstein order O; see also [11].) Take√
3
K = Q( 2), and consider three orders of K with O ⊊ O′ ⊊ OK given by:
√
√
√
√
3
3
3
3
O = Z[2 2, 2 4] = Z + 2 2Z + 2 4Z;
√
√
√
3
3
3
O′ = Z[ 4] = Z + 2 2Z + 4Z;
√
√
√
3
3
3
OK = Z[ 2] = Z + 2Z + 4Z.
√√
We set q = 2Z + 2 3 2Z + 4 3 4Z and note it is a O-ideal of index 4 in O, hence a primary
ideal of O. We show q is not invertible as an O-ideal by contradiction.
If itwere invertible,
√
√
3
3
2
2
then q would also be an invertible O-ideal. Now q = 4Z + 4 2Z + 4 4Z = 4OK hasord(q2 ) = (q2 : q2 ) = OK , so q2 is not invertible for O by Lemma 2.4(2), a contradiction.

RAY CLASSES FOR ORDERS

13

Secondly, q has multiplier ring ord(q) = (q : q) = O. To see this, we observe that the only
orderscontaining O are O′ andOK . It suffices to show qO′ ̸= q, which holds because 2 ∈ q√√
and 3 4 ∈ O′ have product 2 3 4 ∈/ q. Thus, q is not an O′ -ideal, hence ord(q) = O.
There is a factorization theory for invertible integral ideals based on a notion of irreducible integral ideal. In general this theory does not result in unique factorizations, nor are
irreducible integral ideals always prime.
Definition 2.6 (Irreducible integral ideal). An invertible integral ideal q is said to be irreducible for the one-dimensional Noetherian domain D if q ̸= D and the factorization q = abfor invertible ideals a, b ⊆ D implies that a = D or b = D.
Irreducible invertible ideals of one-dimensional Noetherian domains D are necessarily primary ideals. (If they were not primary, they would have a nontrivial primary decomposition,
contradicting irreducibility.) There may be more than one irreducible invertible ideal whoseradical is a given prime ideal, as well as more than one irreducible invertible ideal associatedto a given non-Archimedean valuation on K; both phenomena are illustrated by Example 2.8.
Exampleorders; invertibility). Consider√√ 2.7 (Primary and prime ideals in non-maximal
K = Q( −13), which has ring of integers OK = Z + −13Z of discriminant−52. Let q be√
an inert prime in OK ; for example, q = 5. Then m = qOK = qZ + q −13Z is a maximal
ideal of OK of norm q 2 in OK . It is an invertible√ principal ideal in the maximal order√ OK .
Consider the non-maximal order O = Z + q −13Z. The lattice m = qZ + q −13Z isa maximal O-ideal; hence, it is a prime ideal of O. It has ord(m) = OK , so it is not an
invertible integral ideal of O. Since it is not invertible,√ it cannot be a principal ideal of O.
2
On the other hand, the ideal q := qO = qZ + q −13Z, which has q ⊆ m, is a principal
ideal of O; hence, it is an invertible O-ideal. It√is is a primary ideal of O, and its associatedprime ideal in O is rad(q) = m, noting that (q −13)2 ∈ q.
Example 2.8 (Nonuniquefactorization√√ of an invertible ideal√into irreducible invertible factors). Let K = Q( 2) with OK √= Z+ 2Z, and let O = Z+2 2Z.
√ Then O does not contain2
the fundamental unit ε √= 1 + 2, but does contain√ ε = 3 + 2 2. Now the two O-ideals
q1 = (2ε)O = 4Z+(2+2 2)Z and q2 = 2O = 2Z+4 2Z are principal O-ideals,√ hence invertible. They are both primary ideals associated to the prime ideal p = 2Z+2 2Z = 2OK ⊊ O,
which is not invertible. The ideal p has index 2 in O. Therefore q1 and q2 , which are eachof index 4 in O and index 2 in p, must be irreducible. One has√(q1 )2 = (q2 )2 = 4O = 4Z + 8 2Z.
Thus the invertible ideal 4O has two different irreducible factorizations.
2.3. Conductors and relative conductors of orders of number fields. The conductor
ideal f(O) of an order O of a number field is an important invariant of the order that contains
information on the non-invertible ideals of O.
Definition 2.9. The absolute and relative conductor are defined as follows.
(1) The (absolute) conductor of O (in OK ) isf(O) := fOK(O) = (O : OK ) = {α ∈ OK : αOK ⊆ O}.
It is the largest OK -ideal in O.

14

GENE S. KOPP AND JEFFREY C. LAGARIAS(2) More generally, if O ⊆ O′ , then the relative conductor of O in O′ isfO′(O) = (O : O′ ) = {α ∈ O′ : αO′ ⊆ O}.
It is the largest O′ -ideal in O.
The absolute conductor ideal f(O) = fOK(O) is contained in all relative conductors fO′ (O).
Example 2.10 (Conductors of quadratic orders). If K is ah quadratic
field of discriminant√ i∆, then the maximal order of K is given by OK = O∆ = Z ∆+2 ∆ . The orders of K are of
the form"#p√f 2∆ + f 2∆∆+ ∆Of 2 ∆ = Z
=Z+f
Z
2
2
for f ∈ N. The order Of 2 ∆ has discriminant f 2 ∆. We have Of 2 ∆ ⊆ O(f ′ )2 ∆ if and only if
f ′ |f , and the relative conductor is fO(f ′ )2 ∆ (Of 2 ∆ ) = ff′ O(f ′ )2 ∆ .
In the quadratic field case, the absolute conductor determines the order. This does nothold in general, as the following biquadratic example shows.
Example 2.11 (The absolute conductor does not determine the order). Let K be the field
generated by the 12-th roots of unity, and write it as a biquadratic field K = Q(ω, i), whereω 2 + ω + 1 = 0 and i2 + 1 = 0. The maximal order of K is OK = Z[ω, i].
Consider the two orders O ⊊ O′ ⊊ OK given by
O = Z[5ω, 5i, 5ωi] = Z + 5ωZ + 5iZ + 5ωiZ andO′ = Z[ω, 5i] = Z + ωZ + 5iZ + 5ωiZ.
If α = w + ωx + iy + ωiz ∈ f(O), then α ∈ O =⇒ 5|x, 5|y, 5|z, and iα ∈ O =⇒ 5|w,
so we see that f(O) = 5OK . On the other hand, if α = w + ωx + iy + ωiz ∈ f(O′ ), thenα ∈ O′ =⇒ 5|y, 5|z, and iα ∈ O′ =⇒ 5|w, 5|x, so we see that f(O′ ) = 5OK . Thus, O andO′ have the same absolute conductor 5OK .
Example 2.12 (Noninvertible relative conductor). We compute the relative conductor fororders O and O′ in Example 2.11. Taking α = w + ωx + iy + ωiz ∈ fO′ (O), then α ∈O =⇒ 5|x, 5|y, 5|z, and ωα ∈ O =⇒ 5|(w − x) and thus 5|w; we see that the relative
conductor fO′ (O) = 5OK . This relative conductor is not only an O-ideal, it is also an
OK -ideal. Therefore it is not invertible as an integral O-ideal by Lemma 2.4(2).
The conductor determines all singular prime ideals.
Lemma 2.13. Let O be an order of a number field and p a nonzero prime ideal of O. The
following are equivalent.
(1) p is not coprime to f(O), that is, p + f(O) = p. Equivalently, f(O) ⊆ p.
(2) p is a non-invertible prime ideal of O.
(3) p is a singular prime ideal of O.
Proof. For any prime ideal p we have p ⊆ p + f(O) ⊆ O. Since all nonzero prime ideals p
are maximal, p is not coprime to f(O) if and only if p + f(O) = p.
Proof of (1) ⇔ (2). The contrapositive of this assertion says: p is an invertible prime idealof O if and only if f(O) ̸⊆ p. Since p is maximal, the latter means p + f(O) = O, that is, p
is coprime to f(O). The contrapositive is proved as [46, Ch. I, Prop. 12.10].

RAY CLASSES FOR ORDERS

15

Proof of (2) ⇔ (3). The contrapositive of the assertion says: p is an invertible prime
ideal of O if and only if p is a regular prime ideal of O. Now a prime ideal p is invertible if
and only if pOp = αp Op is a principal ideal, where Op denotes the localization of O at the
maximal ideal p. (See [46, Ch. I, Sec. 11, 12 and Lem. 12.4].) Equivalently, the localization
Op is a discrete valuation ring, so the set of all nonzero ideals in the local ring are powers ofthe maximal ideal pOp , that is, p is regular, as required.
□
In the next example, the norm of an integral ideal a of an order O is the index
NmO (a) := [O : a]with O and a regarded as Z-modules. This definition coincides with the usual definitionof norm of an ideal for the maximal order. Norms are treated in Appendix A, where thedefinition of norm is extended to fractional ideals of an order.
Example 2.14 (Structure of primary ideals for p containing f(O); failure of gcd and lcm).
The set of primary ideals q associated to a prime ideal p containing the conductor ideal f(O)can have a very complicated structure.
(1) The containment of ideals q1 ⊆ q2 does not always imply the divisibility of q1 by q2 .
(2) For suitable pairs of primary ideal, non-existence of gcd and of lcm can occur.
(3) The ideal norm function may not multiplicative on primary ideals.
Let K = Q(i) with maximal order OK = Z[i] = Z + iZ, and consider the order O =Z[2i] = Z + 2iZ of index 2 in OK . The conductor ideal f(O) = 2OK = 2Z + 2iZ is a maximal
ideal of O, since it is of index 2 in O. We consider the set F of all primary ideals associatedto f(O), and sort them by the size of their norm, which is always a power of 2. The uniqueelement of norm 2 in O is Q2 := f(O). Since ord(Q2 ) = OK is larger than O, Q2 is notinvertible for O by Lemma 2.4(2).
There are two invertible ideals in F of norm 4, both principal:
q4 := 2O = 2Z + 4iZ and q′4 := 2iO = 4Z + 2iZ.
There is also one non-invertible ideal of norm 4,
Q4 := 2(1 + i)OK = 4Z + (2 + 2i)Z.
Both q4 and q′4 are contained in Q2 , but they cannot be divisible by Q2 because it is a
non-invertible ideal, while all divisors of invertible ideals are invertible by Lemma 2.4(1).
This illustrates (1).
Note that q4 and q′4 are irreducible integral ideals, as their only (integral ideal) common
divisor is O. However, they are not coprime integral ideals, since q4 + q′4 = Q2 .
In addition Q4 is an irreducible O-integral ideal, because it is not divisible by Q2 viewed
as an O-integral ideal. This holds since (1 + i)OK is not an integral O-ideal, as it contains
1 + i ̸∈ Z[2i]. (Q2 is divisible by (1 + i)OK viewed as an OK -ideal.)
There is a single ideal in F of norm 8, which is Q8 := (Q2 )2 = 4OK = 4Z + 4iZ. It is a
non-invertible ideal, since ord(Q8 ) = OK . Now Q8 = (Q2 )2 , so we have
8 = NmO (Q8 ) = NmO ((Q2 )2 ) > (NmO Q2 )2 = 4.
This illustrates that the ideal norm on O is not always multiplicative, giving assertion (3).
We also note that q4 Q2 = q′4 Q2 = Q8 . The norms are multiplicative in this case, since q4
is invertible; see Proposition A.2 in Appendix A.

16

GENE S. KOPP AND JEFFREY C. LAGARIAS

We now study the primary ideals of norm 16 in F. There are two invertible ideals:
q16 := 4O = 4Z + 8iZ and q′16 := 4iO = 8Z + 4iZ.
There is also a non-invertible ideal of norm 16,
Q16 := 4(1 + i)OK = 8Z + (4 + 4i)Z.
As invertible ideals, q16 and q′16 cannot be divisible by Q2 , Q4 or Q8 . We find thatq16 = (q4 )2 = (q′4 )2

and q′16 = q4 q′4 .

(2.3)

It follows that each of q16 and q′16 are contained in both q4 and in q′4 . In the divisibility partial
order, the factorization (2.3) shows that there are no intermediate elements of the partial
order between q4 (respectively, q′4 ) and either q16 or q′16 (since q4 and q′4 are irreducible).
Consequently, the lcm of q4 and q′4 is not well-defined, and the gcd of q16 and q′16 is not welldefined. This illustrates (2).
The divisiblity partial order relating these ideals is pictured in Figure 1.
q16

q′16

q4

q′4

Figure 1. Hasse diagram for {q4 , q′4 , q16 , q′16 }.
Remark 2.15 (Structure of primary ideals for prime ideals p not containing f(O)). For
prime ideals p coprime to the conductor ideal f(O), none of the pathologies (1)–(3) shown in
Example 2.14 occur. The contrapositive of Lemma 2.13 shows that p is an invertible ideal andis regular, that is, its complete set of p-primary ideals is {pj : j ≥ 1}. For p-primary ideals,
containment is equivalent to divisibility, so gcd and lcm are well-defined. The O-norms of
p-primary ideals are multiplicative by Proposition A.2, because all pj are invertible.
2.4. Monoids of integral ideals coprime to a fixed modulus m. Recall that I(O)denotes the monoid of all integral ideals and I∗ (O) the monoid of all invertible integral
ideals. We introduce monoids of integral ideals and invertible integral ideals coprime to a
fixed modulus m.
Definition 2.16 (Monoids of integral ideals coprime to m). Given an integral ideal m of anorder O of a number field, let Im (O) denote the set of m-coprime integral ideals of O, and
let I∗m (O) denote the set of m-coprime invertible integral ideals. That is,
Im (O) = {a ∈ I(O) : a + m = O},andI∗m (O) = {a ∈ I∗ (O) : a + m = O}.

Lemma 2.17. Let O be an order of a number field and m an integral ideal of O.
(1) The sets Im (O) and I∗m (O) are monoids for O-ideal product.
(2) If m, m′ are ideals of O with m ⊆ m′ , then Im (O) ⊆ Im′ (O), and I∗m (O) ⊆ I∗m′ (O).
Proof of (1). We check that these sets are closed under O-ideal product. If a + m = O andb + m = O, thenO = (a + m)(b + m) = ab + (am + bm + mm) = ab + m.
We noted earlier that invertibility of integral ideals is preserved under O-ideal product. □RAY CLASSES FOR ORDERS

Proof of (2). If a + m = O and m ⊆ m′ , then a + m′ = O, since a + m ⊆ O.

17□

The next result shows that for any m contained in the conductor ideal, the associatedmonoid of integral ideals coprime to m has unique factorization into prime ideals.
Proposition 2.18. Let O be an order of a number field K, with f(O) its absolute conductor
ideal. Suppose the integral ideal m has m ⊆ f(O). Then, the following hold.
(1) The monoid I∗m (O) is a free abelian monoid whose set of generators is the set of prime
ideals P(m) = {p : m ̸⊆ p, p ̸= {0}}. All the prime ideals in P(m) are regular, and all
ideals a ∈ I∗m (O) have unique prime factorization into nonnegative powers of prime
ideals in P(m).
(2) The monoid Im (O) of nonzero ideals coprime to m and the monoid I∗m (O) of invertible
ideals coprime to m coincide.
Proof of (1). We suppose m ⊆ f(O) = (O : OK ). All ideals in Im (O) are coprime to m,hence they are coprime to f(O). But the prime ideals not coprime to f(O) are exactly the
singular prime ideals by Lemma 2.13(3). Thus P(m) contains only regular prime ideals.
Since coprimality to m is preserved by ideal product, all products of a finite number of ideals
drawn from P(m) (allowing multiplicity) belong to Im (O). This exhausts the allowed primary
decompositions of elements of I(O), hence I(O) is a free abelian monoid.
□
Proof of (2). The result follows (1) because all regular prime ideals of O are invertible, byLemma 2.13.
□
Remark 2.19. The conclusions of Proposition 2.18 need not hold for non-maximal orders
when m ̸⊆ f(O). The following examples take m = O.
(1) For some non-maximal orders O, the monoid of all invertible integral ideals I∗ (O) =I∗O (O) is not a free abelian monoid, because its irreducible elements may satisfy′
nontrivial relations. See Example 2.14, (2.3), where q4 and q4 are distinct irreducible′
ideals, but (q4 )2 = (q4 )2 .
(2) For any non-maximal order, the monoid I(O) = IO (O) of all integral ideals contains noninvertible ideals. The conductor ideal has ord(f(O)) = OK ̸= O, so is notinvertible by Lemma 2.4(2).
3. Fractional ideals of orders
This section treats fractional ideals of Noetherian integral domains of dimension one, andin particular, orders of number fields.
The fractional ideal theory of non-maximal orders of number fields has some key differencesfrom that of the maximal order. Not all fractional ideals are invertible, and the set of
fractional ideals under ideal multiplication has the structure of a monoid (a semigroup withidentity) that is not a group. The group of invertible fractional ideals need not be free; it
may contain (a finite number of) torsion elements.
3.1. Fractional ideals for integral domains. Following [18, Defn. 1.1.6], define fractional
ideals for Noetherian integral domains.
Definition 3.1 (Fractional ideal). A fractional ideal a of a Noetherian integral domain D
is a D-submodule of its fraction field K with the property that λa ⊆ D for some λ ∈ K × .
(A integral ideal is a fractional ideal contained in D.)

18

GENE S. KOPP AND JEFFREY C. LAGARIASThe set of fractional ideals of D has sum, product, intersection, and quotient operations.
In particular, the ideal product operation · is defined as in (2.1). The ideal quotient (or colon
ideal ) operation ( : ) is(a : b) := {x ∈ K : xb ⊆ a}.
As for integral ideals, we define the multiplier ring of a fractional ideal a as ord(a) = (a : a).
Proposition 3.2. The set J(D) of all fractional ideals of a Noetherian integral domain D
is closed under the four operations +, ·, ∩, and ( : ) (addition, multiplication, intersection,and ideal quotient.)
Proof. This is [18, Prop. 1.1.11].

□

The set of all fractional ideals of J(D) forms a monoid under the ideal product operation,with identity element D.
We extend the notion of coprimality to that between a fractional ideal and an integral
ideal; we do not define coprimality between two non-integral fractional ideals of D.
Definition 3.3 (Coprimality of a fractional ideal and an integral ideal). A fractional D-ideald is coprime to an integral ideal c ⊆ D if it may be written as a quotient d = (a : b), wherea is an D-integral ideal coprime to c, and b is an invertible D-integral ideal coprime to c.
In this definition, one cannot always choose a, b to be coprime to each other, even when
D = O is an order of a number field; see Example 3.11. Nevertheless one may check that, if
d1 , d2 are coprime to c, then so is their ideal product d1 d2 .
Definition 3.4 (Invertible fractional ideal). A fractional D-ideal a is invertible for D if there
is another fractional ideal b of D such that ab = D. Otherwise it is non-invertible for D.
We let J∗ (D) denote the set of all invertible fractional ideals of D; they form a group undermultiplication in which D is the identity element. There is a converse invertibility result for
fractional ideals that extends Lemma 2.4 for integral ideals.
Lemma 3.5. Let D be a Noetherian integral domain.
(1) If c is an invertible fractional D-ideal, and c = ab with a, b fractional D-ideals, thena and b are both invertible fractional D-ideals.
(2) If a, b, c are all D-fractional ideals (invertible or not) and ord(c) = D, then c = abimplies ord(a) = ord(b) = D. In particular all invertible fractional D-ideals a haveord(a) = D.
Proof. The proof is identical to the proof of Lemma 2.4, replacing O by D and “integral” by“fractional.”□
The next two lemmas show that the definitions of invertibility and coprimality for fractional ideals are consistent with the earlier definitions for integral ideals.
Lemma 3.6. Let D be a Noetherian integral domain, and let a and b be integral D-ideals.
If b is invertible as a fractional ideal of D, then the fractional ideal (a : b) = ab−1 .
Proof. We first show (a : b) ⊆ ab−1 , or equivalently (since b is invertible) that (a : b) b ⊆ a.
Let c ∈ (a : b) and b ∈ b. Thus cb ⊆ a by the definition of (a : b), so in particular, cb ∈ a.
We show the reverse inclusion (a : b) ⊇ ab−1 . Let a ∈ a and d ∈ b−1 . Then, db ⊆ D, so
adb ⊆ aD ⊆ a. By definition of the quotient ideal, ad ∈ (a : b).
□

RAY CLASSES FOR ORDERS

19

Lemma 3.7. Let D be a Noetherian integral domain.
(1) Any invertible fractional ideal a of D with a ⊆ D is an invertible integral ideal, andconversely.
(2) If two integral ideals c, d of D are coprime as integral ideals, then d, regarded as afractional ideal, is coprime to c, and conversely.
Proof of (1). Given an integral ideal a ⊆ D invertible as a fractional ideal, there is a fractional
ideal c such that ac = D. There exists λ ∈ D so that λc ⊆ D is an integral ideal, anda(λc) = λD certifies that a is an invertible integral ideal.
Given an invertible integral ideal a, there is some integral ideal b such that ab = γD fora nonzero γ ∈ D. Then a(γ −1 b) = D certifies a is an invertible fractional ideal.
□
Proof of (2). Suppose c, d are integral ideals satisfying the coprimality condition c + d = D.
Viewing d as a fractional ideal, the decomposition d = (d : D) certifies the fractional ideal dis coprime to c, since d (on the right side of the decomposition) is by hypothesis coprime toc as an integral ideal, and D is an invertible integral ideal coprime to c.
Conversely, we are given integral ideals c, d for which d, regarded as a fractional ideal, iscoprime to c. Then we have d = (a : b) for two integral ideals a, b in which b is invertible
and c + a = c + b = D. By Lemma 3.6, d = ab−1 . Thus, as fractional ideals, we have db = a,and a ⊆ d, since b is an integral ideal. Now c + a = D means there exists c ∈ c and a ∈ asuch that c + a = 1, and necessarily a ∈ d, so c + d = D.
□Invertible fractional ideals are characterized by local data.
Proposition 3.8. If D is any Noetherian integral domain, then a fractional ideal a ∈ J(D) isinvertible for D if and only if each localized ideal am is a principal Dm -ideal for each maximal
ideal m of D.
Proof. This is a special case of Corollary 2.1.7 of [18].

□The next lemma collects basic properties of ideal quotients of fractional ideals and deduces
the inclusions in (5), which justify (1.2), taking D = O and D′ = O′ .
Lemma 3.9. Let a, b, c be fractional ideals in a Noetherian integral domain D.
(1) (a : b) ⊆ (ac : bc), with equality if c is invertible.
(2) (a : b) c ⊆ (ac : b), with equality if c is invertible.
(3) ((a : b) : c) = (a : bc).
(4) If a ⊆ b ⊆ c, then (a : c) ⊆ (a : b) and (a : c) ⊆ (b : c).
Now let m be an integral ideal of D, and suppose D ⊆ D′ with D′ a Noetherian integral
domain that is also a fractional ideal of D. Set fD′ (D) = (D : D′ ). Then
(5) fD′ (D)m ⊆ (m : D′ ) ⊆ fD′ (D) ∩ mD′ .
Proof. Properties (1), (2), (3) and (4) follow directly from the definitions of the quotient andproduct ideals. To prove the left-hand inclusion of property (5), observe using (2) thatfD′ (D)m = (D : D′ ) m ⊆ (Dm : D′ ) = (m : D′ ) .
To prove the right-hand inclusion, use (1) and (4):(m : D′ ) ⊆ (mD′ : D′ D′ ) = (mD′ : D′ ) = mD′ ,and (m : D′ ) ⊆ (D : D′ ) = fD′ (D).

□20GENE S. KOPP AND JEFFREY C. LAGARIAS

3.2. Fractional ideals for orders of number fields. We specialize to the case of an order
O of a number field K.
For the maximal order OK (and more generally for Dedekind domains), the fractional
ideal group J∗ (OK ) is a free abelian group. For non-maximal orders O, J∗ (O) may containa (finite) nontrivial torsion subgroup; see Example 3.11.
Concerning coprimality, each nonzero fractional ideal d of an order O is coprime to all but
a finite set of nonzero prime ideals (i.e., maximal ideals) of O.
Example 3.10 (Non-integral fractional ideals having an ideal power that is an integral
ideal). Consider the non-maximal order O = Z[2i] of the Gaussian field K = Q(i), whose
maximal order is OK = Z[i].
(1) The non-integral fractional ideala := (1 + i)O = 4Z + (1 + i)Z
is a principal fractional ideal; hence, it is an invertible fractional ideal. It has idealsquare a2 = 2iO = q′4 (in the notation of Example 2.14), which is an integral ideal,
shown to be irreducible in Example 2.14.
Thus the irreducible integral ideal q′4 becomes a perfect square viewed in the group
J∗ (O) of invertible fractional ideals. In contrast, the irreducible integral ideal q4 = 2O
is not a perfect square in J∗ (O).
(2) The non-integral fractional ideal
b := (1 + i)OK = 2Z + (1 + i)Z
is a non-invertible fractional O-ideal. It has ideal square b2 = 2OK = Q2 (in thenotation of Example 2.14), which is a non-invertible irreducible integral O-ideal. The
integral ideal Q2 is the square of an element b of the monoid J(O) of fractional ideals.
Example 3.11 (A finite order element of J∗ (O)). Consider the order O = Z + 2iZ, which
is a suborder of OK = Z + iZ in K = Q(i). Its conductor is f(O) = 2OK = 2Z + 2iZ.
The O-fractional ideal d := iO = 2Z + iZ, is not an integral O-ideal since i ̸∈ O. The
ideal d is a torsion element of J∗ (O), with d2 = O. It is an invertible fractional ideal, and
it is another example of a (non-integral) fractional ideal that has an ideal power that is an
integral ideal.
Example 3.12 (An element of J∗ (O) not a quotient of two coprime integral ideals). We
continue to consider the O-fractional ideal d = iO for the non-maximal order O = Z + 2iZof K = Q(i). It is expressible as a ratio iO = ab−1 of two invertible integral O-ideals, taking
a = 2O and b = 2iO. But a + b = 2OK , so a, b are not coprime integral ideals of O.
In fact, iO cannot be expressed as a ratio of two coprime integral O-ideals, which we now
show by contrapositive. Suppose iO = ab−1 for integral O-ideals a and b, and multiply
both sides by b to obtain ib = a. Thus, a = ib ⊆ iO, and a ⊆ O, so a ⊆ O ∩ iO =(Z + 2iZ) ∩ (2Z + iZ) = 2OK . Similarly, b = ia ⊆ iO and b ⊆ O, so b ⊆ 2OK . Thus,a + b ⊆ 2OK , so a and b cannot be coprime.
3.3. Monoids of fractional ideals of orders coprime to a modulus m. Recall that
J(O) is the monoid of fractional ideals of the order O, and J∗ (O) is the group of invertible
fractional ideals of O. We have introduced a notion of m-coprimality for fractional ideals in
Definition 3.3. We now define monoids of fractional ideals and groups of invertible fractional
ideals coprime to a modulus m which is an integral ideal.

RAY CLASSES FOR ORDERS

21

Definition 3.13 (Monoids of fractional ideals coprime to m). Given an integral ideal m ofan order O of a number field, let Jm (O) denote the set of m-coprime fractional ideals of O,
Jm (O) = {a ∈ J(O) : a is coprime to m}.
Let J∗m (O) denote the set of m-coprime invertible fractional ideals,J∗m (O) = {a ∈ J∗ (O) : a is coprime to m}.
Lemma 3.14. Let O be an order of a number field and m an integral O-ideal.
(1) The set Jm (O) of m-coprime fractional ideals is a monoid under O-fractional ideal
product.
(2) The set J∗m (O) of m-coprime invertible fractional ideals is a group under O-fractional
ideal product.
(3) If m, m′ are ideals of O with m ⊆ m′ , then Jm (O) ⊆ Jm′ (O), and J∗m (O) ⊆ J∗m′ (O).
Proof of (1) and (2). Let c1 , c2 ∈ Jm (O); we check that c1 c2 ∈ Jm (O). Write ci = ai b−1foriintegral ideals ai , bi coprime to m. Then c1 c2 = (a1 a2 )(b1 b2 )−1 . The ideals a1 a2 and b1 b2 are
coprime to m by Lemma 2.17(1), so c1 c2 ∈ Jm (O). Moreover, if c1 and c2 are invertible (thatis, in J∗m (O)), then so is c1 c2 . Both Jm (O) and J∗m (O) have identity element O, and J∗m (O)has inverses (ab−1 )−1 = ba−1 .
□Proof of (3). The result is inherited from the inclusions in Lemma 2.17(2) applied to integral
ideals a, b defining an element c = ab−1 ∈ Jm (O).
□The next result shows that for all ideals m contained in the conductor ideal, the associatedmonoid of integral ideals coprime to m have unique factorization into prime ideals.
Proposition 3.15. Let O be an order of a number field K. Suppose the integral ideal m has
m ⊆ f(O). Then the following hold.
(1) The monoid J∗m (O) of invertible fractional ideals is a free abelian group whose generators are the set of prime ideals P(m) = {p : m ̸⊆ p, p ̸= 0}. All of these prime ideals
are regular. Ideals a ∈ J∗m (O) have unique factorization into integral powers of theprime ideals in P(m).
(2) The monoid Jm (O) of fractional ideals coprime to m under ideal multiplication andthe monoid J∗m (O) of invertible fractional ideals coprime to m coincide.
Proof of (1). This result follows from the unique factorization property of Proposition 2.18(1)using the fact that all p ∈ P(m) are invertible integral ideals, as shown in Proposition 2.18(2),hence invertible fractional ideals.
□
Proof of (2). This follows from the fact that all regular prime O-ideals are invertible.

□

Remark 3.16. The conclusions of Proposition 3.15 for fractional ideals do not generally
hold for m ̸⊆ f(O), e.g., for m = O and O non-maximal.
(1) For some non-maximal orders O, choosing m = O, the monoid of all invertible
fractional ideals J∗ (O) = J∗O (O) is not a free abelian monoid, because it contains
nontrivial torsion elements. See Example 3.11.
(2) For any non-maximal order O, the monoid J(O) = JO (O) of all fractional ideals
contains non-invertible ideals. The conductor ideal f(O) is a noninvertible fractional
ideal, because its non-invertibility as an integral ideal (by Lemma 2.13) implies thesame as a fractional ideal by Lemma 3.7(1).

22GENE S. KOPP AND JEFFREY C. LAGARIAS

4. Change of orders in a number field: extension and contraction of ideals
For two orders O ⊆ O′ having the same quotient field K, we consider the effect of natural
maps sending (integral and fractional) O-ideals to O′ -ideals, and vice versa.
Definition 4.1. The inclusion map O ,→ O′ defines extension and contraction maps on
integral ideals.
(1) If a is an integral ideal of O, then the extension ext(a) := aO′ is the integral O′ -ideal
generated by the elements of a.
(2) If a′ is an integral ideal of O′ , then the contraction con(a′ ) := a′ ∩ O is the integral
O-ideal of elements of a′ also in O.
This is a special case of extension and contraction of ideals under ring homomorphism [4, p.
9]. In Section 4.1, we study the effect of ext and con on general integral ideals. In Section 4.2,
we show these maps are bijective monoid homomorphisms when restricted to submonoids ofintegral ideals coprime to the relative conductor fO′ (O). In Section 4.3, we extend ext andcon to bijective homomorphisms on groups of invertible fractional ideals coprime to fO′ (O).
4.1. Extension and contraction of general integral ideals. Given the inclusion of twoorders ι : O → O′ of a fixed algebraic number field K, we have well-defined functions
ext : I(O) → I(O′ ) and con : I(O′ ) → I(O).
The extension map ext : I(O) → I(O′ ) is a monoid homomorphism for ideal multiplication.
It is easy to check that this map preserves invertibility of ideals, and it preserves the property
of being a principal ideal. The map ext may not be surjective; see Example 4.4.
The contraction map con : I(O′ ) → I(O) in general is not a monoid homomorphism.
One always has con(a′ ) con(b′ ) ⊆ con(a′ b′ ), but strict inclusion may sometimes hold; see
Example 4.5. Contraction need not preserve invertibility of ideals nor the property of being a principal ideal; see Example 4.6. We will show in Section 4.2 that con is a monoid
homomorphism (and thus preserves invertibility) when restricted to If (O′ ), where f = fO′(O).
For later use, we study the effect of ext and con on maximal ideals.
Lemma 4.2. Let O ⊆ O′ be orders of a number field K, and let con and ext be the contraction and extension maps on integral ideals.
(1) If p is a maximal ideal of O, then con(ext(p)) = p.
(2) If p′ is a maximal ideal of O′ , then p = con(p′ ) is a maximal ideal of O.
(3) If p is a maximal ideal of O, then p = con(p′ ) for some maximal ideal p′ of O′ .
(4) If p′ is a maximal ideal of O′ , then ext(con(p′ )) ⊆ p′ , and strict inequality may occur.
To prove Lemma 4.2, we will need a standard result in commutative algebra.
Lemma 4.3. Let A ⊆ B and C be commutative rings with unity such that C is the integral
closure of A in B. Let a be an ideal of A, and let ext(a) the extension of a to C. Let a be
the integral closure of a in B, that is,
a = {α ∈ C : f (α) = 0 for a monic polynomial f (x) with all coefficients in a}.
Then, a = rad(ext(a)), so in particular a is a B-ideal.
Proof. This is [4, Lem. 5.14].

□

Proof of Lemma 4.2(1). The inclusion a ⊆ con(ext(a)) holds for all a ∈ I(O) [4, Prop.
I.17]. Since p is maximal, either con(ext(p)) = O or con(ext(p)) = p. Suppose for a

RAY CLASSES FOR ORDERS

23

contradiction that con(ext(p)) = O. Then, ext(p) contains O, so it contains O′ O = O′ , soext(p) = O′ . It follows that the extension to the maximal order extOK(p) = OK . But OK
is the integral closure of O in K, so by Lemma 4.3, p = rad(OK ) = OK . Thus, 1 ∈ p,
so f (1) = 0 for some monic polynomial f (x) = xn + µn−1 xn−1 + · · · + µ0 with µj ∈ p. So
1 = − (µn−1 + · · · + µ0 ) ∈ p, which means p = O, contradicting the hypothesis that p is a
maximal ideal of O. It follows that con(ext(p)) = p.
□
Proof of Lemma 4.2(2). Consider any a ∈ O such that a ∈/ con(p′ ). Then, p′ + aO′ is an
ideal of O′ strictly containing p′ , so p′ + aO′ = O′ . Thus, a is invertible in O′ /p′ , and O′ /p′
is finite (indeed, a finite field), so there is some n ∈ N such that an ≡ 1 (mod p′ ). That is,an = 1 + p′ for some p′ ∈ p′ . However, p′ = an − 1 ∈ O, so in fact an ≡ 1 (mod con(p′ )).
Thus, con(p′ )+aO = O. Since this holds for any a ∈ O \con(p′ ), we have shown that con(p′ )is a maximal ideal.
□
Proof of Lemma 4.2(3). By (1) we have con(ext(p)) = p. Let p′ be any maximal idealcontaining ext(p). By maximality of p, either con(p′ ) = p or con(p′ ) = O. The latter case
implies 1 ∈ p′ , a contradiction.
□
Proof of Lemma 4.2(4). The inclusion ext(con(a)) ⊆ a holds for general ideals in I(O′ ) [4,Prop. I.17]. An example of strict inclusion is given in Example 4.4 below.
□′
Example 4.4. [For a maximal
O-ideal p, ext(p)Consider√√ need not be√a maximal O -ideal.]2′′
the orders O = Z + 5 2Z and O = Z + 5 2Z in K = Q( 2),√ with O ⊆ O . Both these
2Z. Consider ext : I(O) →orders are strictly smaller than the maximal order
O
=
Z
+
K
√
′
′
2
I(O ) and con : I(O ) → I(O). Set p = 5Z + 5 2Z. By inspection p is an integral ideal ofO, and since it is of prime index 5 in O, it is a maximal ideal of O. By Lemma 4.3(3)√there
is a maximal ideal p′ of O′ such that con(p′ ) = p. We may in fact choose p′ = 5Z + 5 2Z.
On the other hand, p = 5O′ is by inspection a principal O′ -ideal. It follows that ext(p) = p.
Now p ⊊ p′ ⊊ O′ , so p = ext(con(p′ )) is not maximal as an O′ -ideal.
The maximal ideal p′ of O′ is not in the image of the map ext: Suppose it were, sop′ = ext(a) for some a ∈ I(O). Then, ext(con(ext(a))) = ext(a) by [4, Prop. I.17]. But

ext(con(ext(a))) = ext(con(p′ )) = ext(p) = p,so we would obtain p = ext(a) = p′ , which is false.
Example 4.5. [The contraction map between ideal monoids I(O′ ) and I(O) need√ not be2
a monoid homomorphism.]
consider the orders O = Z + 5 2Z and√√As in Example 4.4,
O′ = Z + 5 2Z in K = Q( √2), with O ⊆ O′ . We√consider con : I(O′ ) → I(O). Also, as in
Example 4.4, let p = 5Z + 52 2Z and p′ = 5Z + 5 2Z; we have con(p′ ) = p.
Take a′ = b′ = p′ . As an O′ -ideal,

√ 2
√
′ ′
′ 2
a b = (p ) = 5Z + 5 2Z = 52 Z + 52 2Z.
Now a′ b′ = (p′ )2 is also an O-ideal, so that√con(a′ b′ ) = con((p′ )2 ) = (p′ )2 = 52 Z + 52 2Z.
√
Since con(p′ ) = p = (5Z + 52 2Z), we have√con(a′ ) con(b′ ) = (con(p))2 = (p2 ) = 52 Z + 53 2Z.
We have shown con(a′ ) con(b′ ) ⊊ con(a′ b′ ).

24

GENE S. KOPP AND JEFFREY C. LAGARIAS

Example4.6. [Contraction and extension√of ideals in non-maximal orders] Consider K =√Q( −13)√ with maximal order OK = Z + −13Z, and consider a non-maximal order O =Z + q −13Z where q is anfrom Example 2.7√ odd inert prime (for example, q = 5). Recall
2that m = qOK = qZ + q −13Z is a maximal OK -ideal of norm q , and m is principal in
OK , hence invertible.
Now m is also an O-ideal and is a maximal ideal for O. It is the conductor ideal for O,
so it is non-invertible as an O-ideal. It is immediate that ext(m) = m. For the contraction
map con : OK → O, we havecon(m) = m ∩ O = m.
This example verifies con(ext(m)) = m, and also ext(con(m)) = m, preserving the maximal
ideal property. However, the contracted ideal m is not an invertible fractional O-ideal hence
also not a principal O-ideal.
√
Furthermore, consider q := qO = qZ + q 2 −13Z. It is a principal O-ideal, so it is aninvertible O-ideal; hence, it cannot be an OK -ideal. We have q ⊊ m. It is easy to see thatext(q) = m. So we now have con(ext(q)) = con(m) = m and q ⊊ con(ext(q)) = m. (Thuscon(ext(a)) can be a maximal ideal while a is not maximal.)
4.2. Extension and contraction of integral ideals coprime to the relative conductor. Extension and contraction operations behave well when restricted to integral ideals
coprime to the relative conductor ideal.
Lemma 4.7. On the set of integral ideals If (O) coprime to the (relative) conductor f =
fO′ (O) ∈ I(O) ∩ I(O′ ), contraction con : If (O′ ) → If (O) defines an isomorphism of monoids,with inverse the extension map ext : If (O) → If (O′ ).
Proof. We first show that con and ext are bijections inverse to each other. For general ringmaps, it is easily seen that ext(con(a′ )) ⊆ a′ and a ⊆ con(ext(a)) [4, Prop. I.17]. The reverse
inclusions are not true in general, even in our case of the inclusion map of a suborder in anorder. We must use coprimality to the conductor.
For the first, consider a′ ∈ If (O′ ), and set a = con(a′ ) = a′ ∩O. We will show ext(con(a′ )) =a′ . By comprimality of a′ to the relative conductor f = fO′ (O) we have
1 = a′ + c ′(4.1)

for some a′ ∈ a′ and c ∈ f ⊆ O. Then a′ = 1 − c ∈ O, so a′ ∈ a = con(a′ ). Then (4.1)
certifies that a is coprime to f in the order O. Also a′ ∈ ext(con(a′ )), so ext(con(a′ )) is
coprime to f. We must show a′ ⊆ ext(con(a′ )). We have a′ f ⊆ con(a), because a′ f ⊆ f ⊆ O
and a′ f ⊆ a′ O′ = a′ . Now, given b′ ∈ a′ , we have from (4.1) thatb′ = b′ a + b′ c ′ .
Now b′ a ∈ ext(con(a)) since b′ ∈ O and a ∈ con(a), while b′ c′ ∈ a′ f ⊆ con(a) ⊆ ext(con(a)),
hence b′ ∈ ext(con(a′ )). Thus, a′ ⊆ ext(con(a′ )), so we conclude that ext(con(a′ )) = a′ .
For the second, consider a ∈ If (O), and set a′ = ext(a). We will show con(ext(a)) = a.
By the coprimality assumption, there exist a ∈ a and f ∈ f = fO′ (O) with 1 = a + f . Sincea ∈ a′ = ext(a), the ideals a′ and f are coprime in O′ , and in addition a ∈ con(ext(a)). We
must show con(ext(a)) ⊆ a. Suppose b ∈ con(ext(a)) ⊆ O; then the coprimality equation
implies b = ba + bf . We show b ∈ a by showing both summands of the right hand side arein a. Now b ∈ O, so ba ∈ Oa = a. Now
bf ∈ ext(a)f = (aO′ )f = a(O′ f) = af ⊆ aO = a.

RAY CLASSES FOR ORDERS

25

Thus con(ext(a)) ⊆ a, whence con(ext(a)) = a.
Finally, for two ideals a, b ∈ If (O), it follows from the definition of the extension map thatext(ab) = ext(a) ext(b). Because con defines an inverse to ext, it follows that con is also ahomomorphism from If (O′ ) onto If (O).
□4.3. Extension and contraction of fractional ideals coprime to the relative conductor. The extension and contraction maps between orders O ⊆ O′ of a number field K
consistently extend from integral ideals to fractional ideals, provided that one restricts to
fractional ideals coprime to the relative conductor fO′ (O).
Proposition 4.8. Consider two orders O ⊆ O′ of the number field K. Let f = fO′ (O) denote
the relative conductor. Let m′ be an integral ideal of O′ having m′ ⊆ f. Then the contractionand extension maps extend uniquely to isomorphisms between groups of fractional ideals
coprime to m′ . That is, the maps
con : Jm′ (O′ ) → Jm′ (O) andext : Jm′ (O) → Jm′ (O′ )are well-defined and are inverses of each other.
Remark 4.9. The extension map on fractional ideals a coprime to fO′ (O) is ext(a) = aO′ ,as in the case for integral ideals in Definition 4.1(1). However, the contraction map on
fractional ideals coprime to fO′ (O) requires a new definition different from Definition 4.1(2)for integral ideals. That is, the contraction map on fractional ideals does not always havecon(a′ ) = a′ ∩ O, although the inclusion a′ ∩ O ⊆ con(a′ ) holds. For example, let b′ be any
proper integral O′ -ideal coprime to fO′ (O), and set a′ = (b′ )−1 . Then O′ ⊊ a′ , so a′ ∩ O = O.
However, by Proposition 4.8, we will have con(a′ ) = con(b′ )−1 = (b ∩ O)−1 = b−1 = a′ . Sinceb′ is a proper ideal, con(a′ ) ̸= a′ ∩ O.
Proof of Proposition 4.8. Note that m′ is both an integral O-ideal and an integral O′ -ideal,
the latter by assumption, and the former because O ⊆ O′ (so m′ is a fractional O-ideal) andm′ ⊆ fO′ (O) ⊆ O, hence m′ is integral as an O-ideal by Lemma 3.7(1).
We first claim that the maps con : Im′ (O′ ) → Im′ (O) and ext : Im′ (O) → Im′ (O′ ) send
integral ideals in the specified domains into integral ideals in the specified codomains.
To prove the claim for con, suppose a′ ∈ Im′ (O′ ), so a′ + m′ = O′ . Then there exist a′ ∈ a′and m′ ∈ m′ such that a′ + m′ = 1. Now m′ ∈ m′ ⊆ f ⊆ O, so a′ = 1 − m′ ∈ O, showing thata′ ∈ con(a′ ). Therefore con(a′ ) is coprime to m′ in O.
To prove the claim for ext, suppose a ∈ Im′ (O), so a + m′ = O. Then, there exist a ∈ aand m ∈ m′ with a + m = 1. Clearly, a ∈ ext(a). Therefore, ext(a) is coprime to m′ in O′ .
Now Lemma 4.7 asserts that con(ext(a)) = a and ext(con(a′ )) = a′ for all integral ideals
in their respective domains. Because the domains and codomains of the maps above matchon integral ideals, the isomorphisms given by Lemma 4.7 restricts to bijective isomorphisms
con : Im′ (O′ ) → Im′ (O) and ext : Im′ (O) → Im′ (O′ ) that are inverses of each other.
We now consider any fractional ideal d = (a : b) = ab−1 ∈ Jm′ (O) with a ∈ Im′ (O)and b ∈ I∗m′ (O), where d = ab−1 by Lemma 3.6. We define ext(d) = ext(a) ext(b)−1 ;any group homomorphism extending ext must be defined thus. To show this definition is
independent of the choice of expression of d as a ratio of integral ideals, consider two such−1
expressions d = a1 b−1
1 = a2 b2 . Then, a1 b2 = a2 b1 , so ext(a1 ) ext(b2 ) = ext(a2 ) ext(b1 ), so−1
ext(a1 ) ext(b1 ) = ext(a2 ) ext(b2 )−1 , whence ext(d) is well-defined. By a similar argument,

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GENE S. KOPP AND JEFFREY C. LAGARIAS

defining con(a′ (b′ )−1 ) = con(a′ ) con(b′ )−1 for a′ ∈ Im′ (O′ ) and b′ ∈ I∗m′ (O′ ) gives a uniquewell-defined homomorphism. The fact that con and ext are inverses of each other then follows
from the same fact for integral ideals.
□
5. Ray class groups of orders
In this section, we define ray class groups of an order O as quotients of certain groups offractional ideals, and we show that those groups can be taken to satisfy auxiliary coprimenessconditions to an arbitrary ideal d.
5.1. Definition of ray class groups of orders. Let O be an order of a number field K.
Definition 5.1. A fractional ideal a of O is principal if it may be written as a = αO forsome α ∈ K. The group of principal fractional ideals is denoted by P(O).
When working with ray class groups with level datum (O; m, Σ), one needs to talk about
modular congruences between non-integral elements of K. This requires a bit of care, givenin the following definition.
Definition 5.2. Given α, β ∈ K, we say α ≡O β (mod m) (abbreviated α ≡ β (mod m)when O is known from context) if α = αα21 and β = ββ12 for some α1 , α2 , β1 , β2 ∈ O with
α2 O and β2 O coprime to m, satisfying α1 β2 − α2 β1 ∈ m. This is equivalent to saying thatα − β ∈ mO[Sm−1 ], where O[Sm−1 ] is the semilocal ring obtained by inverting the elements ofO coprime to m. (See also Definition 5.6.)
Definition 5.3 (Principal ray ideal group). Given an integral ideal m in O and a subset Σof the real places of K (possibly empty), define the group of principal ray ideals of O modulo(m, Σ), denoted Pm,Σ (O), by:Pm,Σ (O) = {αO : α ∈ K × such that α ≡ 1 (mod m) and ρ(α) > 0 for all ρ ∈ Σ}.
Definition 5.4. The ray class group of O modulo (m, Σ) is
Clm,Σ (O) =J∗m (O).
Pm,Σ (O)That is,Clm,Σ (O) ={invertible fractional ideals of O coprime to m}.
{αO : α ∈ K × such that α ≡ 1 (mod m) and ρ(α) > 0 for all ρ ∈ Σ}This definition of the ray class group for an order O parallels the definition of the ray classgroup for the maximal order. To make the definition flexible, we will show in Section 5.3
that one may add auxiliary congruence conditions without changing the group, for example,
requiring coprimality to the conductor ideal f(O).
Definition 5.5. The (wide) ring class group (or Picard group) Cl(O) of an order O is the
special case of modulus (O, ∅), so that
Cl(O) := ClO,∅ (O) =J∗O (O){invertible fractional ideals of O}=.
PO,∅ (O){αO: α ∈ K × }One can show this definition of ring class group (and the corresponding ring class field
described by splitting of degree one prime ideals) is consistent with the classical definitionsin the case of quadratic fields, as given in [16, pp. 114–115] and [17, pp. 162–163].

RAY CLASSES FOR ORDERS

27

5.2. Local behavior of ideals. Before proving results about the ray class group, we collectsome facts about the interaction of localization with invertibility and ideal inclusion.
We introduce a notation for localization of rings, for later use.
Definition 5.6. For a commutative ring with unity R and an ideal I of R, we denote by SI
the multiplicatively closed set of elements coprime to I,
SI := {a ∈ R : aR + I = R}.
We denote by R[SI−1 ] the ring defined by inverting the elements of SI . (We avoid the notation
SI−1 R to prevent any confusion with multiplication of ideals.)
If R is a Noetherian ring of dimension 1 (such as an order in a number field), then forany nonzero ideal I, the ring R[SI−1 ] is a semilocal ring—a ring with finitely many maximal−1
ideals. For example, the ring Z[S(6)] consists of those rational numbers whose denominators
are contain no factors of 2 or 3, and the only maximal ideals are (2) and (3). If p is a maximal
ideal of R, then R[Sp−1 ] = R[(R \ p)−1 ] = Rp is termed the localization away from p.
The localization of an R-modules M may be defined by the extension of scalars M [SI−1 ] :=M ⊗R R[SI−1 ] (or by an equivalent direct construction given in [4, Ch. 3]). We use the notation
Mp := M [Sp−1 ] = M ⊗R Rp for the localization of an O-module M away from a prime ideal
p. In the cases we consider, the natural map M → Mp is injective, so we will drop the tensor
product notation, simply writing M [Sp−1 ] = M Rp = Rp M .
The following proposition recalls a basic fact from commutative algebra: Fractional idealcontainment (and more generally, injectivity of module maps) is a local property.
Proposition 5.7. For any commutative ring R and a map of R-modules ϕ : M → N , thefollowing are equivalent:(1) ϕ is injective.
(2) The induced map ϕp : Mp → Np is injective for every prime ideal p of R.
(3) The induced map ϕp : Mp → Np is injective for every maximal ideal p of R.
In particular, for an order O and fractional ideals a, b ∈ J(O),a ⊆ b ⇐⇒ ap ⊆ bp for all nonzero prime ideals p ⊆ O.
Proof. The statement for R-modules is [4, Prop. 3.9]. The statement for fractional O-ideals
follows by taking ϕ : a → b to be the inclusion map.
□
We next recall a sharpening of Proposition 3.8, valid for orders of number fields.
Proposition 5.8. Let O be an order in a number field K. Invertible fractional ideals of O
are locally principal: If a ∈ J∗ (O), then ap = aOp is a principal ideal of Op . Moreover, thecorrespondence a 7→ (ap ) := (aOp ) defines an isomorphism
M
J∗ (O) ∼P(Op ),=
pwhere p varies over the nonzero prime ideals of O, and P(Op ) = {αOp : α ∈ Kp } is thegroup of nonzero principal fractional ideals of Op in its quotient field Kp .
Proof. This is [46, Ch. I, Prop. 12.6] and is also proved in [55, Thm. 2.14].

□

We now show that the group of invertible fractional ideals coprime to m is determined by
the set of nonzero prime ideals containing m.

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Lemma 5.9. Let O be an order in a number field K, and let m1 , m2 be nonzero integral
ideals of O. Then:(1) One has {p : p a prime O-ideal with m1 ⊆ p} ⊆ {p : p a prime O-ideal with m2 ⊆ p}
if and only if Im2 (O) ⊆ Im1 (O).
(2) If Im2 (O) ⊆ Im1 (O), then J∗m2 (O) ⊆ J∗m1 (O).
e ∈ I∗ (O) such that
(3) For any nonzero O-ideal m, there exists an invertible ideal m
e ⊆ m and Im (O) = Ime (O) (and thus J∗m (O) = J∗me (O)).
m
Proof. If m1 ⊆ p but m2 ̸⊆ p, then p ∈ Im2 (O) (because p is maximal and hence p + m2 = O)
but p ∈/ Im1 (O) (because p + m1 = p ̸= O). This proves the “if” direction of (1) (by proving
the contrapositive). To prove the “only if” direction of (1), suppose{p1 , . . . , pm } = {nonzero prime ideals p ⊇ m1 } ⊆ {nonzero prime ideals p ⊇ m2 }.
For any integral ideal b ∈ I(O), we haveb + m2 = O =⇒ bOpj + m2 Opj = Opj for 1 ≤ j ≤ m=⇒ bOpj = Opj for 1 ≤ j ≤ m, because m2 Opj ⊆ pj Opj
and pj Opj is the unique maximal ideal of Opj=⇒ bOp + m1 Op = Op for all nonzero prime O-ideals p,
because m1 Op = Op for p ∈/ {p1 , . . . , pm }=⇒ b + m1 = O by Proposition 5.7.
In other words, Im2 (O) ⊆ Im1 (O).
To prove (2), by definition J∗m (O) consists of fractional ideals of the form ab−1 wherea, b ∈ I∗m (O). Thus, Im2 (O) ⊆ Im1 (O) =⇒ I∗m2 (O) ⊆ I∗m1 (O) =⇒ J∗m2 (O) ⊆ J∗m1 (O).
To prove (3), consider a nonzero O-ideal m. Let {p1 , . . . , pm } be the set of all prime ideals
containing m. For each j with 1 ≤ j ≤ m, choose some nonzero element µj ∈ mOpj . Bye ∈ J∗ (O) such that mO
e pj = µj Opj
Proposition 5.8, there exists a unique invertible ideal m
e p = Op for p ∈for 1 ≤ j ≤ m and mO/ {p1 , . . . , pm }. We havee ⊆ p}.
{nonzero prime ideals p with m ⊆ p} = {nonzero prime ideals p with mThus, by (1), both Ime (O) ⊆ Im (O) and Im (O) ⊆ Ime (O). Then, by (2), J∗m (O) = J∗me (O).

□5.3. Auxiliary coprimality conditions on ray class groups of orders. In the definitionof the ray class group for the maximal order OK , it is well known that restricting the fractional
ideal group to fractional ideals coprime to some ideal d, and restricting the principal ideals
similarly, yields the same group. In this subsection we show this is true for arbitrary orders.
Being able to impose an auxiliary coprimeness condition to some ideal d on the fractional
ideal groups, done in Lemma 5.12, makes possible the comparison of ray class groups fordifferent orders and different ray class moduli, particularly the comparison of arbitrary rayclass groups with certain ray class groups of the maximal order.
Definition 5.10 (Principal ray ideal group coprime to d). Given an integral O-ideal d, thegroup of principal ray ideals (modulo m) coprime to d, denoted Pdm,Σ (O), is given by:
Pdm,Σ (O) = {αO : α ∈ K × , α ≡ 1 (mod m), αO coprime to d, ρ(α) > 0 for all ρ ∈ Σ}.
The following lemma shows that, without loss of generality, we may suppose d ⊆ m.

RAY CLASSES FOR ORDERS

29

dm
Lemma 5.11. If d is an integral O-ideal, then Pdm,Σ (O) = Pd∩mm,Σ (O) = Pm,Σ (O). In particum
lar, Pm,Σ (O) = Pm,Σ (O).
d∩mdProof. Clearly Pdm
m,Σ (O) ⊆ Pm,Σ (O) ⊆ Pm,Σ (O). To show the reverse inclusions, supposea ∈ Pdm,Σ (O). Write a = αO for α ∈ K × such that α ≡ 1 (mod m), α coprime to d, andρ(α) > 0 for all ρ ∈ Σ. Write α = αα12 for α1 , α2 ∈ O with α2 coprime to m (which is possible
by Definition 5.2). Then α1 ≡ α2 (mod m), so α1 is also coprime to m, so α is coprime tom. Since α is coprime to d and to m, α is also coprime to dm and to d ∩ m.
The equality Pm,Σ (O) = Pm□m,Σ (O) follows by taking d = O.

The following key lemma is the most technically tricky result in this paper.
Lemma 5.12. For any nonzero ideals d ⊆ m ⊆ O and any set Σ of real places of K (possibly
empty),J∗ (O)
Clm,Σ (O) = dd.
(5.1)
Pm,Σ (O)Proof. Consider the inclusion J∗d (O) ,→ J∗m (O), and compose with the quotient map to get ahomomorphism
J∗ (O)= Clm,Σ (O).
(5.2)
ϕ : J∗d (O) → m
Pm,Σ (O)The kernel ker ϕ = J∗d (O) ∩ Pm,Σ (O) = Pdm,Σ (O). For the equality (5.1), it suffices to provethat ϕ is surjective.
Given b ∈ J∗m (O), to show surjectivity we must find some βO ∈ Pm,Σ (O) (equivalently,
β −1 O ∈ Pm,Σ (O)) such that a = β −1 b ∈ J∗d (O). To construct a suitable element β, theargument will move to the maximal order OK and then back to O.
e ⊆ m with
For the integral ideal m, by Lemma 5.9(3), there exists some invertible ideal me ⊆ qj is
the property that the set {q1 , . . . qn } of all nonzero prime ideals of O for which m
identical to the set of nonzero prime ideals with m ⊆ qj . By Proposition 5.8, the invertiblee b are locally principal: For each nonzero prime ideal p of O,
fractional ideals m,
e p = µp Op for some µp ∈ K × , and we may choose µp = 1 whenever p ∈(1) mO/ {q1 , . . . , qn };(2) bOp = βp Op for some βp ∈ K × , and we may choose βp = 1 whenever b ̸⊆ p.
The finite set of prime ideals p with b ̸⊆ p is a union of two subsets:(1) a subset (possibly empty) of the set of prime ideals {pi : 1 ≤ i ≤ m} having d ⊆ pi
and m ̸⊆ pi ;(2) a set (possibly empty) of prime ideals {rk : 1 ≤ j ≤ ℓ} having d ̸⊆ rk .
(Note that the prime ideals pi , qj , rk need not be invertible.)
The condition that b+m = O is equivalent to the condition that b ̸∈ qj for 1 ≤ j ≤ n, and,e = O. Additionally, it follows that the sets {p1 , . . . , pm },in turn, to the condition that b + m{q1 , . . . , qn }, and {r1 , . . . , rℓ } are all disjoint.
To show surjectivity (5.2) our object is to multiply b by an element in Pm,Σ (O) to force
an additional coprimality condition with respect to the prime ideals {p1 , · · · pm } withoutchanging its behavior locally at the prime ideals qi . The set of primes {r1 , . . . , rℓ }, being
disjoint from both the set of pi and the set of qj , play no role in the following argument.
We now move to the maximal order OK . For 1 ≤ i ≤ m, write pi = con(p′i ) for somenonzero prime ideal p′i of OK ; this is possible by Lemma 4.2. Similarly, for 1 ≤ j ≤ n,write qj = con(q′j ) for some nonzero prime ideal q′j of OK . These primes p′i , q′j are all

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GENE S. KOPP AND JEFFREY C. LAGARIAS

distinct, because the primes pi , qj are all distinct. Thus, there are pairwise independent
multiplicative valuations (absolute values) |·|v1 , . . . , |·|vm , |·|w1 , . . . , |·|wn on K corresponding
to p′1 , . . . , p′m , q′1 , . . . q′n , respectively.
Let fp be the “local conductor” at p, that is,
fp = {x ∈ Op : xOp ⊆ Op },where Op is the integral closure of Op in its fraction field Kp = K. Let f˜p be any nonzero
element of fp , so f˜p Op is a nonzero principal ideal contained in fp .
By the Approximation Theorem [46, Thm. 3.4], we can find β ∈ K such that(1) |β − βpi |vi < |β|vi for 1 ≤ i ≤ m,(2) |β − 1|wj ≤ f˜qj µqj−1

for 1 ≤ j ≤ n, andwj(3) ρ(β) > 0 for ρ ∈ Σ, via the archimedean bound |β − 1|ρ < 1.
We now define the invertible fractional O-ideal a by a := β −1 b.
We have aOp = β −1 βp Op for all prime O-ideals p. By (1), for 1 ≤ i ≤ m, |β −1 βpi − 1|vi < 1,×so in particular, β −1 βpi is a unit in Op′i = Opi . But Opi ∩ Opi = Op×i , so β −1 βpi is a unit
in Opi , and thus aOpi = Opi . On the other hand, for 1 ≤ j ≤ n, aOqj = β −1 Oqj becauseqj ⊇ m, and thus βqj = 1. It follows from (2) that |β −1 − 1|wj < 1, so in particular, β −1
is a unit in Oq′j = Oqj and thus a unit in Oqj , so aOqj = Oqj . Since aOp = Op for allp ∈ {p1 , . . . , pm } ∪ {q1 , . . . , qn }, it follows that a is coprime to d, so a ∈ J∗d (O).
−1by (2), so β −1 − 1 ∈ f˜q µq Oq′ . We have
Moreover, for 1 ≤ j ≤ n, |β −1 − 1| ≤ f˜q µq
wj

j

j

wj

j

j

j

the inclusion of ideals
e qj ,
f˜qj µqj Oq′j = µqj (f˜qj Oqj ) ⊆ µqj Oqj = mOe qj for each j. It follows that β −1 ≡ 1 (mod m),e so in particular, β −1 ≡so β −1 − 1 ∈ mOe ⊆ m. Combining this congruence with the positivity condition (3), we
1 (mod m), since m−1−1
have ab = β O ∈ Pm,Σ (OK ).
□5.4. Effect of change of order and modulus on ray class groups of orders. While
it is clear that Lemma 5.12 implies the existence of surjective change-of-modulus maps
Clm,Σ (O) → Clm′ ,Σ′ (O) whenever m ⊆ m′ and Σ ⊇ Σ′ , it is also true (but not immediately
obvious) that one can change the order O. For orders O ⊆ O′ , the change-of-order map
between ray class groups is induced by the extension map ext : J∗ (O) → J∗ (O′ ) on fractional
ideals. We thus call it the induced extension map ext.
We show that the induced extension map is well-defined and surjective. This result will
be applied and refined in Proposition 6.6 to explicitly describe the kernel of ext.
Lemma 5.13. Let K be a number field, and consider level data L = (O; m, Σ) and L′ =(O′ ; m′ , Σ′ ) for K such that O ⊆ O′ , mO′ ⊆ m′ , and Σ ⊇ Σ′ . There exists a unique
homomorphismL′ext = extL : Clm,Σ (O) → Clm′ ,Σ′ (O′ )satisfying ext([a]) = [ext(a)] for a ∈ J∗m (O), and this map is surjective. For another level datum L′′ = (O′′ , m′′ , Σ′′ ) for K such that O′ ⊆ O′′ , m′ O′′ ⊆ m′′ , and Σ′ ⊇ Σ′′ , the compatibility
L′′
L′
L′′
relation extL′ ◦ extL = extL holds.

RAY CLASSES FOR ORDERS

31

Proof. Uniqueness follows from the formula ext([a]) = [ext(a)], provided that this formuladefines a well-defined homomorphism.
To see that ext is a well-defined, it suffices to show the image of Pm,Σ (O) under ringextension from O to O′ is contained in Pm′ ,Σ′ (O′ ). So consider an ideal b ∈ Pm,Σ (O). Thenb = bO for some b ∈ K, b − 1 ∈ mO[Sm−1 ], and ρ(b) > 0 for ρ ∈ Σ. Since mO[Sm−1 ] ⊆mO′ [Sm−1 ] ⊆ m′ O′ [Sm−1′ ], we have b − 1 ∈ m′ O′ [Sm−1′ ], that is, b ≡ 1 (mod m′ ). Additionally,
since Σ ⊇ Σ′ , we have ρ(b) > 0 for ρ ∈ Σ′ . Thus, ext(b) ∈ Pm′ ,Σ′ (O′ ).
Let f = fO′ (O) be the relative conductor of O in O′ . Let d be an ideal of O′ containedin both m and f. (For example, take d = (m : O′ ).) Using Lemma 5.12, under the inclusion
map on ideals, we haveJ∗m′ (O′ )J∗d (O′ )′Clm′ ,Σ′ (O ) :== d.
Pm′ ,Σ′ (O′ )Pm′ ,Σ′ (O′ )To see that the map ext is surjective, let A ∈ Clm′ ,Σ′ (O′ ), and let a ∈ J∗d (O′ ) be a representative of the class A. Because a is coprime to the conductor f, we have ext(con(a)) = a by
Lemma 4.7, so ext([con(a)]) = [a] = A.
L′′
L′
L′′
The compatibility relation extL′ ◦ extL = extL is a direct consequence of the formulaext([a]) = [ext(a)].
□6. Exact sequences for ray class groups of orders
In this section, we compare ray class groups for pairs of level data for the same field. So
as to have a convenient shorthand to indicate which pairs of level data are comparable, we
introduce a partial order on level data.
Definition 6.1. Let L = (O; m, Σ) and L′ = (O′ ; m′ , Σ′ ) be level data for the same number
field K. (That is, O, O′ are K-orders, m is an integral O-ideal, m′ is an integral O′ -ideal,and Σ, Σ′ ⊆ {embeddings K ,→ R}.) We say that
L ≤ L′
if and only if
O ⊆ O′ , mO′ ⊆ m′ , and Σ ⊇ Σ′ .
L′

When L ≤ L′ , the induced extension map extL is shown in Lemma 5.13 to be a surjective
group homomorphism from Clm,Σ (O) to Clm′ ,Σ′ (O′ ). In this section, we completely describe
the kernel of ext in terms of local and global unit groups.
6.1. Exact sequences relating unit groups and principal ideals for varying orders.
We describe unit groups that we will relate by exact sequences to groups of principal idealsof varying orders.
Definition 6.2. For a commutative ring with unity R and an ideal I of R, define the group
UI (R) := {α ∈ R× : α ≡ 1 (mod I)} = (1 + I) ∩ R× .
If R has real embeddings, and Σ is a subset of the real embeddings of R, define
UI,Σ (R) := {α ∈ R× : α ≡ 1 (mod I) and ρ(α) > 0 for ρ ∈ Σ}.
We also make use of the following extension of this notation: If there is an obvious map
ϕ : R1 → R2 implicit in the discussion, and if I is an ideal of R1 , then we will let UI (R2 ) :=Uϕ(I)R2 (R2 ) and UI,Σ (R2 ) := Uϕ(I)R2 ,Σ (R2 ).

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GENE S. KOPP AND JEFFREY C. LAGARIAS

Proposition 6.3. For any level datum (O; m, Σ), we have an exact sequence
1 → Um,Σ (O) → Um,Σ (O[Sd−1 ]) → Pdm,Σ (O) → 1,
where Sd = {α ∈ O : αO + d = O}.
Proof. By definition,
Pdm,Σ (O) = {αO : α ∈ O[Sd−1 ], α ≡ 1 (mod m) , ρ(α) > 0 for all ρ ∈ Σ},
so ϕ(α) := αO defines a surjective map ϕ : Um,Σ (O[Sd−1 ]) → Pdm,Σ (O). Moreover,ker(ϕ) = {α ∈ O[Sd−1 ] : αO = O, α ≡ 1 (mod m) , ρ(α) > 0 for all ρ ∈ Σ}
= Um,Σ (O).
The proposition follows.

□

We now relate unit groups and principal ideal groups of varying orders.
Proposition 6.4. Let K be a number field, and consider level data L = (O; m, Σ) andL′ = (O′ ; m′ , Σ′ ) for K such that L ≤ L′ . Let d be any O′ -ideal such that d ⊆ (m : O′ ). Then
we have a short exact sequence of the form
1→

Pdm′ ,Σ′ (O′ )Um′ ,Σ′ (O′ )Um′ (O′ /d)′→× {±1}|Σ\Σ | → d→ 1.
Um,Σ (O)Um (O/d)Pm,Σ (O)Proof. Here d is an integral O-ideal because d ⊆ (m : O′ ) = {a ∈ K : aO′ ⊆ m} ⊆ m ⊆ O,
and dO ⊆ dO′ = d. Thus O/d is well-defined. We localize away from d by inverting Sd , forthe two rings O and O′ separately. By Proposition 6.3, we have short exact sequences
Um,Σ (O) → Um,Σ (O[Sd−1 ]) → Pdm,Σ (O) → 1↓↓↓
1 → Um′ ,Σ′ (O′ ) → Um′ ,Σ′ (O′ [Sd−1 ]) → Pdm′ ,Σ′ (O′ ) → 1.
1 →The downward maps are all injective—in particular, the rightmost map is—so, by the snake
lemma, the sequence of cokernels is exact:
1→Pdm′ ,Σ′ (O′ )Um′ ,Σ′ (O′ [Sd−1 ])Um′ ,Σ′ (O′ )→→→ 1.
Um,Σ (O)Pdm,Σ (O)Um,Σ (O[Sd−1 ])This is the short exact sequence in the proposition statement, except for the middle group.
We now must show the middle group is isomorphic to the group in the proposition statement.
First of all, note that the localization maps induce compatible isomorphisms∼

Um (O/d) −→ Um (O[Sd−1 ]/dO[Sd−1 ])↓↓∼ι′ : Um′ (O′ /d) −→ Um′ (O′ [Sd−1 ]/dO′ [Sd−1 ]).
ι:Let Υ = {embeddings K ,→ R}. Consider the map′

ϕ : Um′ ,Σ′ (O′ [Sd−1 ]) → Um′ (O′ /d) × {±1}Υ\Σ
given by ϕ(u) = ι′ −1 u mod dO′ [Sd−1 ] , (sign(ρ(u)))ρ∈Υ\Σ′ .
′We see that ϕ is surjective as follows: Consider (u, ε) ∈ Um′ (O′ /d) × {±1}Υ\Σ . Choose alift ue ∈ O′ of u; ue ∈ Um′ (O′ [Sd−1 ]) because it is coprime to d and congruent to 1 modulo m′ .
We may replace ue with u
e + λ for any λ ∈ d ∩ m′ = d without affecting its image in Um′ (O′ /d).
Since d forms a lattice in K ⊗ R, we may choose λ appropriately so that sign(ρ(eu + λ)) = +1

RAY CLASSES FOR ORDERS

33

for ρ ∈ Σ′ and sign(ρ(eu + λ)) = ερ for ρ ∈ Υ \ Σ. Thus, ue + λ ∈ Um′ ,Σ′ (O′ [Sd−1 ]), andϕ(eu + λ) = (u, ε).
Now, define′
Um′ (O′ /d) × {±1}Υ\Σ′ −1
ϕ : Um′ ,Σ′ (O [Sd ]) →Um (O/d) × {±1}Υ\Σ
by ϕ(u) := ϕ(u) mod Um (O/d) × {±1}E\Σ ; ϕ is surjective because ϕ is. Furthermore, we
compute the kernel of ϕ as follows.
−1u mod dO′ [Sd−1 ] ∈ Um (O/d) and sign(ρ(u)) = 1 for all ρ ∈ Σ \ Σ′ .
ϕ(u) = 1 ⇐⇒ ι′
We already knew that sign(ρ(u)) = 1 for all ρ ∈ Σ′ , so in fact, the second condition tells usthat ρ(u) > 0 for all ρ ∈ Σ. We now reformulate the first condition.
−1
ι′ u mod dO′ [Sd−1 ] ∈ Um (O/d)
⇐⇒ u mod dO′ [Sd−1 ] ∈ ι′ (Um (O/d)) = ι(Um (O/d)) = Um O[Sd−1 ]/dO[Sd−1 ]⇐⇒ u ≡ 1 mod mO[Sd−1 ] .
This last condition also implies that u ∈ O[Sd−1 ] (rather than simply being in O′ [Sd−1 ]), sinced is an integral O-ideal. Thus, we have shown that
ker ϕ = {u ∈ Um′ ,Σ′ (O′ [Sd−1 ]) : u ∈ O[Sd−1 ], u ≡ 1 mod mO[Sd−1 ] , ρ(u) > 0 for ρ ∈ Σ}
= Um,Σ (O[Sd−1 ]).
Therefore, ϕ induces an isomorphism′

Um′ ,Σ′ (O′ [Sd−1 ]) ∼ Um′ (O′ /d) × {±1}E\Σ ∼ Um′ (O′ /d)|Σ\Σ′ |
×
{±1}
,
−
→
=
−1
Um (O/d) × {±1}E\ΣUm (O/d)Um,Σ (O[Sd ])proving the proposition.

□

6.2. Exact sequences relating ray class groups of varying orders. We relate classgroups of varying orders and moduli; the formula (6.1) is important for applications. The following results generalize results presented in [46, Ch. 1, Sec. 10] corresponding to the special
case L = (O; O, ∅), L′ = (OK ; OK , ∅). More losely speaking, they generalize results presented
in [16, Sec. 7.2] corresponding to the special case L = (OK ; m, Σ), L′ = (OK ; OK , ∅).
Theorem 6.5. Let K be a number field, and consider level data L = (O; m, Σ) and L′ =(O′ ; m′ , Σ′ ) for K such that L ≤ L′ in the sense of Definition 6.1. Let d be any O′ -ideal suchthat d ⊆ (m : O′ ). With the U-groups defined as in Definition 6.2, we have the following
exact sequence.
Um′ ,Σ′ (O′ )Um′ (O′ /d)′1→→
× {±1}|Σ\Σ | → Clm,Σ (O) → Clm′ ,Σ′ (O′ ) → 1.
(6.1)
Um,Σ (O)Um (O/d)To prove this result, we use the following proposition.
Proposition 6.6. Let K be a number field, and consider level data L = (O; m, Σ) andL′ = (O′ ; m′ , Σ′ ) for K such that L ≤ L′ . Let d be any O′ -ideal such that d ⊆ (m : O′ ). Then
we have an exact sequence of the form
1→

Pdm′ ,Σ′ (O′ )→ Clm,Σ (O) → Clm′ ,Σ′ (O′ ) → 1.
Pdm,Σ (O)34

GENE S. KOPP AND JEFFREY C. LAGARIAS

Proof. The extension map ext : Clm,Σ (O) → Clm′ ,Σ′ (O′ ) is surjective by Lemma 5.13, so thesequence is exact at Clm′ ,Σ′ (O′ ).
Note that d is both an O′ -ideal and O-ideal, because it is an O′ -ideal and d ⊆ (m : O′ ) ⊆m ⊆ O. In addition d ⊆ m, so that d ⊆ m′ (because m ⊆ m′ ). By Lemma 5.12,
Clm,Σ (O) =Id (O)Id (O′ )
′
′
′
and
Cl
(O
)
=
.
m ,Σ
Pdm,Σ (O)Pdm′ ,Σ′ (O′ )The kernel of the extension map isker(ext) = {[a] ∈ Clm,Σ (O) : aO′ = αO′ , α ∈ O′ , α ≡ 1 (mod m′ ) , ρ(α) > 0 for all ρ ∈ Σ′ }.
The ideal d ⊆ (O : O′ ) = fO′ (O), so fractional ideals (of O or O′ ) coprime to d are also
coprime to the conductor fO′ (O). By Proposition 4.8, con and ext act as inverses on ideals
coprime to the conductor, so setting ϕ(αO′ ) = [con(αO′ )] defines a surjective map
ϕ : Pdm′ ,Σ′ (O′ ) → ker(ext).
Moreover,dker(ϕ) = {αO′ : α ∈ O[Sd−1 ], α ≡ 1 (mod m) , ρ(α) > 0 for ρ ∈ Σ} = Pd∩mm,Σ (O) = Pm,Σ (O).

The proposition follows.

□

We now prove the main exact sequence.
Proof of Theorem 6.5. The result follows by gluing together the two short exact sequencesin Proposition 6.4 and Proposition 6.6.
□
We also give the specialization of Theorem 6.5 to L = (O; m, Σ) and L′ = (O; O, ∅). Thiscase will be important for the application to SIC-POVMs [1, 34].
Corollary 6.7. Let K be a number field and O an order of K. Let m be an ideal of O andΣ ⊆ {embeddings K ,→ R}. With the U-groups defined as in Definition 6.2, we have thefollowing exact sequence.
1→

O×→ (O/m)× × {±1}|Σ| → Clm,Σ (O) → Cl(O) → 1.
Um,Σ (O)In particular, the kernel×|Σ|(O/m) × {±1}ker(Clm,Σ (O) ↠ Cl(O)) ∼=im(O× ),

where im(O× ) is the image of global units under the map ε 7→ (ε, (sgn(ρ(ε)))ρ∈Σ ).
Proof. Take ray class level data L = (O; m, Σ) and L′ = (O; O, ∅), and choose d = m in
Theorem 6.5. All the hypotheses are satisfied, noting d = m = (m : O). The terms in theexact sequence simplify, with Um (O/m) = Ud (O/d) being the trivial group, UO (O/d) =(O/m)× , and UO,∅ (O) = O× .
□Remark 6.8. Corollary 6.7 parallels a result of Campagna and Pengo [10, Thm. 4.6] for
their idèlic formulation of class field theory for orders. The kernel ker(Clm,Σ (O) ↠ Cl(O)) isO
isomorphic to the Galois group Gal Hm,Σ/H O by Theorem 1.3, so Corollary 6.7 describes
the structure of this Galois group. More generally, Theorem 6.5 describes the structure of′O
the Galois group Gal Hm,Σ/HmO′ ,Σ′ .

RAY CLASSES FOR ORDERS

35

6.3. Cardinality of ray class groups of orders. The following result gives a formulafor the “class number” of the ray class group with level datum (O; m, Σ). It generalizes aformula given by Neukirch [46, Thm. I.12.12] for the cardinality of the Picard group of anorder (corresponding to level datum (O; O, ∅)) as well as a formula given by Cohn [16, Thm.
7.2.7] for the “unit ray class number” (corresponding to level datum (OK ; m, Σ)).
Theorem 6.9. Let K be an algebraic number field having r real places and s conjugate pairsof complex places. Let OK be the maximal order, and let (O; m, Σ) be a level datum of K.
O×The groups Um,ΣK(O) and Clm,Σ (O) are finite, and one has
2|Σ| #(OK /(m : OK ))×hK·# Clm,Σ (O) = ×# Um (O/(m : OK ))[OK : Um,Σ (O)]

(6.2)

where hK is the class number of K. In particular, one has×rank(Um,Σ (O)) = rank(OK) = r + s − 1.

(6.3)

Proof. We specialize the short exact sequence in Theorem 6.5, given m ⊆ O, for the level
data L = (O; m, Σ) and L′ = (OK ; OK , ∅). We choose d = (m : OK ). All the hypotheses ofTheorem 6.5 are satisfied, since mOK ⊆ OK = m′ , and we obtain
1→

×OK(OK /(m : OK ))×→× {±1}|Σ| → Clm,Σ (O) → Cl(OK ) → 1.
Um,Σ (O)Um (O/(m : OK ))(6.4)

The second nontrivial term in this exact sequence is clearly finite, and the fourth term is finite
of order hK by the finiteness of the class group. It follows that the other two terms are finite.
Equation (6.3) follows from the finiteness of the first term and Dirichlet’s Unit Theorem.
Moreover, the alternating product of the cardinality of the terms in an exact sequence offinite groups is 1. Writing this product and solving for # Clm,Σ (O) yields (6.2).
□
6.4. Ring class groups of orders. Theorem 6.5 allows us to express the (wide) ring classgroup Cl(O) = ClO,∅ (O) of an order O, as defined in Definition 5.5, as a quotient of the
Takagi ray class group Clf (Ok ) = Clf(O),∅ (OK ) (of the maximal order) for the conductor idealf(O), permitting us to quantify the difference between them.
To understand the structure of the (wide) ring class group Cl(O), take L = (O; O, ∅) and′L = (OK ; OK , ∅). Now choose d = f = f(O). In particular, since d = (O : OK ), this is
the case already used in (6.4) with the further specialization (m, Σ) = (O, ∅). Theorem 6.5
applies to give the exact sequence
1→

×OK(OK /f)×→→ Cl(O) → Cl(OK ) → 1.
O×(O/f)×For the Takagi ray class group, choose instead L = (OK ; f(O), ∅) and L′ = (OK ; OK , ∅).
Again set d = f = f(O), and note d ⊆ (f(O) : OK ) = f(O). Note that the unit group
Uf (OK /f) = {1}, so we obtain the exact sequence
1→

×OK→ (OK /f)× → Clf (OK ) → Cl(OK ) → 1.
Uf (OK )There are natural quotient maps making the following diagram commute (because Uf (OK )is a subgroup of O× ), so there is a surjective induced map ψ from the ray class group to the36

GENE S. KOPP AND JEFFREY C. LAGARIAS

ring class group.
×OK
Uf (OK )1(OK /f)×

Clf (OK )Cl(OK )

1

Cl(OK )

1

ψ
×
OK
O×

1

(OK /f)×(O/f)×Cl(O)In the next section, we generalize this comparison by introducing a ray class modulus.

7. Ray class fields of orders
In this section, we define and prove existence of class fields associated to the ray classO
groups of Section 5. As usual, we fix a number field K. We will attach a ray class field Hm,Σto the level datum L = (O; m, Σ) for K; this field will be an abelian extension of K.

7.1. Ray class fields of orders defined via Takagi ray class groups. We define rayclass fields of orders by the following recipe. We are given the level datum (O; m, Σ), and
we also consider the level datum (OK ; (m : OK ), Σ). We define a homomorphism ψ from thegroup Cl(m:OK ),Σ (OK ) to the given ray class group Clm,Σ (O) and show that it is surjective.
We let J = J(O; m, Σ) = ker ψ be the kernel. As a subgroup of Cl(m:OK ),Σ (OK ), J has an
associated Takagi ray class (sub)field L = LJ (of H(m:OK ),Σ ), given in Theorem 7.2(1) below.
O
assigned to the level
We define the field L obtained this way to be the ray class field Hm,Σdatum (O; m, Σ) for the order O.
To construct ψ, we start from the exact sequence from Theorem 6.5 for L = (O; m, Σ) and′L = (OK ; OK , ∅) with and d := (m : OK ). It is
1→

×OK(OK / (m : OK )))×→× {±1}|Σ| → Clm,Σ (O) → Cl(OK ) → 1,
Um,Σ (O)Um (O/ (m : OK ))since Um′ (O′ /d) = (OK / (m : OK ))× . Consider also a second exact sequence given by
Theorem 6.5, taking L = (OK ; (m : OK ), Σ), L′ = (OK ; OK , ∅), and d = (m : OK ). In
this exact sequence, the denominator in the second term is U(m:OK ) (OK / (m : OK )) (since((m : OK ) : OK ) = (m : OK )), which is the trivial group. We obtain
1→

×
OK

U(m:OK ),Σ (OK )→ (OK / (m : OK ))× × {±1}|Σ| → Cl(m:OK ),Σ (OK ) → Cl(OK ) → 1.

As in Section 6.3, there are natural quotient maps between the objects in these two exact
sequences corresponding to the downward maps labeled κ, π, and id in the following diagram.
The map κ exists and is surjective because U(m:OK ),Σ (OK ) is a subgroup of Um,Σ (O); themap π is a quotient in the first coordinate; and the map id is the identity map. Moreover,
it is straightforward to check that the diagram commutes, which implies that there is an

RAY CLASSES FOR ORDERS

37

induced surjective map ψ in the position shown.
1

×
OK

U(m:O ),Σ (OK )
K
κ

1

×
OK
Um,Σ (O)(OK / (m : OK ))× × {±1}|Σ|π(OK /(m:OK ))×× {±1}|Σ|Um (O/(m:OK ))Cl(m:OK ),Σ (OK )ψ

Clm,Σ (O)Cl(OK )1

id

Cl(OK )

1
(7.1)

We have thus constructed a surjective map ψ : Cl(m:OK ),Σ (OK ) ↠ Clm,Σ (O).
We make the following definition.
O
Definition 7.1. The ray class field of the order O with modulus (m, Σ) is the subfield Hm,ΣofOK
the Takagi ray class field H(m:OK ),Σ associated to J(O; m, Σ) := ker ψ in (7.1) under the GaloisOK
correspondence between subgroups of the Galois group Gal(H(m:O/K) ∼= Cl(m:OK ),Σ (OK )K ),Σ
OKand subfields of H(m:O
containing K.
K ),Σ
O
Theorem 1.1 will identify Hm,Σin terms of data associated to the splitting of primes in
OK , their contractions to O, and a ray class congruence condition on those contractions.
O
produced by this definition is the unique extension
Namely, we will show the field L = Hm,Σfield of K whose set of prime ideals p over OK that split completely in L/K agrees (withsymmetric difference a finite set) with the set of prime ideals p of OK whose contractionp ∩ O to O is a principal prime ideal πO having a generator π ≡O 1 (mod m) and ρ(π) > 0
for ρ ∈ Σ.

7.2. The classical existence theorem. We state the classical existence theorem of classfield theory, called the Weber–Hilbert–Artin–Takagi [WHAT] correspondence by Cohn [16,Chap. 7], in our notation.
Theorem 7.2 (WHAT correspondence). Let K be a number field, m an ideal of OK , and Σa subset of the set of real embeddings of K.
(1) (Weber–Takagi) Let J be a subgroup of the (Takagi) ray class group Clm,Σ (OK ). Then,
there is a unique abelian extension LJ /K with the property that a prime ideal p of
OK splits completely in LJ if and only if the ray class [p] lies in J, with finitely many
exceptions. (The exceptions are among the prime ideals dividing m.)(a) If J1 ⊆ J2 , then LJ2 ⊆ LJ1 .
(b) For J = {I}, where I = [OK ] is the principal ray class modulo (m, Σ), the field
OK
L = L{I} = Hm,Σis the principal ray class field.
OK(2) (Artin) Under the correspondence (1), for L = L{I} = Hm,Σthere is an isomorphism
Art : Clm,Σ (OK ) → Gal(L/K), the so-called Artin isomorphism Art
h =i Artm,Σ , which

is determined by sending prime ideals [p] of OK (coprime to m) to L/K∈ Gal(L/K),pand extending this map multiplicatively to all ray ideals [a] coprime to m. Under thisisomorphism, LJ is the fixed field of the principal ray class field L = L{I} under theaction of the group of automorphisms Art(J) ⊆ Gal(L/K); that is,
OK Art(J)
LJ := Hm,Σ.

38

GENE S. KOPP AND JEFFREY C. LAGARIASihdenotes the Frobenius automorphism σp ∈In the statement of (2), the Artin symbol L/Kp
Gal(L/K) computed for a prime ideal P of OL lying over p as σp (x) ≡ xq (mod P) for allx ∈ OL , where q = pj is the number of elements in the finite field OK /p.
Proof. This statement (1) is extracted from [16, Thm. 7.4.1], whereas (2) follows from [16,Thm. 7.4.2]. See also [27, Thm. 6.6.8] for (1) and [27, Thm. and Defn. 6.6.2] for (2).
□OK
Remark 7.3. For J a subgroup of Clm,Σ (OK ), with L = Hm,Σand LJ = LArt(J) as in
Theorem 7.2, one can also define an Artin isomorphism ArtJ making the diagram
Artm,Σ

Clm,Σ (OK )Clm,Σ (OK )/J

ArtJ

Gal(L/K)

Gal(LJ /K)

commute, as a direct consequence of the Galois correspondence. Moreover, the Artin maps
with different class field moduli are compatible because of their local definition via the
Frobenius map. That is, if ϕ : Clm,Σ (OK ) → Clm′ ,Σ′ (OK ) is the natural quotient map, thenArtker ϕ = Artm′ ,Σ′ .
7.3. Proof of Theorem 1.1. The main step in the proof is to identify the map ψ introducedin Section 7.1 with contraction to O on the set of fractional ideals of OK coprime to (m : OK )(or equivalently, coprime to mOK ∩ f(OK )).
Lemma 7.4. For a class [a] ∈ Cl(m:OK ),Σ (OK ) represented by some a ∈ J(m:OK ),Σ (OK ),the map ψ : Cl(m:OK ),Σ (OK ) → Clm,Σ (O) defined by (7.1) may be explicitly written ψ([a]) =[con(a)].
Proof. We apply Proposition 4.8 for the case O ⊆ OK , taking m′ := (m : OK ) and noting that (m : OK ) ⊆ f(O); see Lemma 3.9. Proposition 4.8 constructed an isomorphism
con : J(m:OK ) (OK ) → J(m:OK ) (O) and showed that its inverse map was ext : J(m:OK ) (O) →J(m:OK ) (OK ). For a principal ideal a ∈ P(m:OK ),Σ (OK ) with a = αOK , note that a = ext(αO);
thus, con(a) = con(ext(αO)) = αO. Thus, the composition
J(m:OK ) (OK )con

J(m:OK ) (O)Jm (O)

sends the subgroup P(m:OK ),Σ (OK ) to a subgroup of Pm,Σ (O) and thus defines a map
ψe : Cl(m:OK ),Σ (OK ) → Clm,Σ (O).
To show that ψe = ψ, it suffices to show that ψe makes the diagram in (7.1) commute. On
the left, for a pair (α, ε) ∈ (OK /((m : OK )))× × {±1}|Σ| , the square looks like(α, ε){βOK : β ≡ α (mod (m : OK )) and ρ(β) = ερ }eψ(α (mod Um (O/((m : OK )))) , ε){βO : β ≡ α (mod m) and ρ(β) = ερ },and we observe that it commutes. On the right, ψe clearly does not change the class of a in
Cl(OK ), so the right square commutes as well. So ψe = ψ, and the lemma is proved.
□

RAY CLASSES FOR ORDERS

39

Proof of Theorem 1.1. Consider the map ψ : Cl(m:OK ),Σ (OK ) → Clm,Σ (O) defined by (7.1),
Art(J)O
and let J = ker ψ. By Theorem 7.2, Hm,Σ:= H(m:OK ),Σ
is the unique abelian extension
Oof K such that a prime p of OK splits completely in Hm,Σ if and only if [p] lies in J. But[p] ∈ J if and only if ψ([p]) = 0, and by Lemma 7.4, ψ([p]) = [con(p)] = [p ∩ O]. (Since p isa maximal ideal, p ∩ O is also a maximal ideal by Lemma 4.2(2).)□
7.4. Proof of Theorem 1.2. We prove a more general result.
Theorem 7.5. For two orders O ⊆ O′ in a number field K and any level datum (O; m, Σ),O′OO′
there are inclusions of ray class fields HmO′ ,Σ ⊆ Hm,Σ ⊆ H(m:O ′ ),Σ .
Proof. Consider the following (commutative) diagram, with the dotted lines denoting maps
induced by the others. In this diagram, rows are exact, but the columns are not exact; allvertical maps are surjective.
1(O′ )×U(m:O′ ),Σ (O′ )(O′ / (m : O′ ))× × {±1}|Σ|Cl(m:O′ ),Σ (O′ )Cl(O′ )

1

Cl(O′ )

1 (7.2)Cl(O′ )

1ψ

1(O′ /(m:O′ ))×× {±1}|Σ|Um (O/(m:O′ ))(O′ )×Um,Σ (O)Clm,Σ (O)ϕ1(O′ )×
UmO′ ,Σ (O′ )(O′ /mO′ )× × {±1}|Σ|ClmO′ ,Σ (O′ )The horizontal rows are exact sequences from Theorem 6.5 with data given in the following
table.
Theorem 6.5
top row
middle row
bottom row

L = (O; m, Σ) L′ = (O′ ; m′ , Σ′ )d(O′ ; (m : O′ ) , Σ)(O′ ; O′ , ∅)(m : O′ )′′(O; m, Σ)(O ; O , ∅)(m : O′ )(O′ ; mO′ , Σ)(O′ ; O′ , ∅)mO′

In the second nontrivial column, we have used in all rows that UO′ (O′ /d) = (O′ /d)× and in
the top and bottom rows row that Ud (O/d) = {1}.
The vertical maps in the first two nontrivial columns are given by quotienting by everythingthat is 1 modulo m (from the top row to the middle) and then by everything that is 1
modulo mO′ (from the middle row to the bottom). These maps are well-defined because(m : O′ ) ⊆ m ⊆ mO′ . The commutativity of the leftmost two squares is thus clear. The
commutativity of the diagram, excepting the dotted lines, follows by exactness because, on
the longer horizontal rectangles, it simply encodes an equality between two zero maps.
The maps denoted by dotted lines are induced, so the whole diagram (7.2) commutes.
The upper induced map ψ was described in Section 7.1. The lower induced map ϕ is equal
to the map induced by extension of ideals, as can be seen by commutativity of the diagramand comparison with the “change of order” exact sequence
1

UmO′ ,Σ (O′ )Um,Σ (O)UmO′(O′ /(m:O′ ))Um(O/(m:O′ ))Clm,Σ (O)ϕClmO′ ,Σ (O′ )1obtained from Theorem 6.5 (taking L = (O; m, Σ), L′ = (O′ ; mO′ , Σ), and d := (m : O′ )).

40

GENE S. KOPP AND JEFFREY C. LAGARIAS

Similarly, the composition ϕ ◦ ψ fits into the “change of modulus” exact sequence
1

UmO′ ,Σ (O′ )U(m:O′ ),Σ (O′ )UmO′ (O′ / (m : O′ ))Cl(m:O′ ),Σ (O′ )ϕ◦ψ

ClmO′ ,Σ (O′ )1,also a special case of Theorem 6.5 (where we take L = (O′ , (m : O′ ) , Σ), L′ = (O′ , mO′ , Σ),and d := (m : O′ )).
The diagram (7.2) establishes (from its third nontrivial column) the sequence of surjections
Cl(m:O′ ),Σ (O′ )ψClm,Σ (O)ϕClmO′ ,Σ (O′ ).

Now by Theorem 7.2(1), we have the tower of ray class fields of orders
′

O
H(m:O
′ ),Σ
ker ψ
O
Hm,Σ
ker ϕ
′

O
HmO
′ ,Σ

K
with Galois groups as labelled, thus proving the theorem.

□

Proof of Theorem 1.2. This is the special case O′ = OK of Theorem 7.5.

□

7.5. Proof of Theorem 1.3. We now prove Theorem 1.3, giving a form of Artin reciprocity
for a ray class group and ray class field of an order. The result is obtained from the usual
Artin reciprocity law together with properties of the map ψ in (7.1) established earlier inthis section.
OK
Proof of Theorem 1.3. Let H1 = H(m:O, and let Art : Cl(m:OK ),Σ (OK ) → Gal(H1 /K) beK ),Σ
the (usual) Artin map. Let ψ : Cl(m:OK ),Σ (OK ) → Clm,Σ (O) be the map constructed in (7.1).
By Lemma 7.4, for any class [b] ∈ Cl(m:OK ),Σ (OK ), one has ψ([b]) = [con(b)].
O
Let H0 = Hm,Σ. By Definition 7.1 and the Galois correspondence it describes, there is an
isomorphism ArtO making the following diagram commute.

Cl(m:OK ),Σ (OK )Art∼

Gal(H1 /K)
(7.3)

ψ

Clm,Σ (O)ArtO∼

Gal(H0 /K)

To give an explicit formula for ArtO , consider any fractional ideal a of O coprime tof(O) ∩ m (or, equivalently, coprime to (m : OK )). We compute ArtO ([a]) by lifting [a] toCl(m:OK ),Σ (OK ) along ψ. The coprimality condition implies that con(aOK ) = con(ext(a)) = a(by Proposition 4.8), so ψ([aOK ]) = [con(ext(a))] = [a], that is, [aOK ] is a lift of [a]. Therefore (7.3) gives
ArtO ([a]) = Art([aOK ])|H0 .

RAY CLASSES FOR ORDERS41

Now let p be any prime ideal of O coprime to f(O) ∩ m, with residue field O/p having
characteristic p. Let P be a prime ideal of OH0 lying over pOK . We claim that for allα ∈ OH0 ,ArtO ([p])(α) ≡ αq (mod P) ,(7.4)
where q = pj is the number of elements in O/p, identifying ArtO ([p]) as a Frobenius automorphism.
To see this, let P be a prime ideal of OH1 lying over P. By Artin reciprocity (part (2) of
Theorem 7.2; see also Remark 7.3), for any α ∈ OH0 ,ArtO ([p])(α) = Art([pOK ])|H0 (α)
Now the condition that p is coprime to f(O) ∩ m implies that pOK is a prime ideal of OK by
Lemma 4.2 and Proposition 4.8; in addition, P and P are unramified over pOK because H0
and H1 only have ramification over K at the primes dividing f(O) ∩ m (that is, the primes
dividing (m : OK )). Now for any β ∈ OH1 we have by definition′

Art([pOK ])(β)) ≡ β q (mod P) ,where q ′ is the number of elements in OK /pOK . We have q ′ = q since the hypotheses
guarantee that O/p ∼= OK /pOK . Applying this congruence with β = α ∈ OH0 , noting thatq
P ∩ OH0 = P and α ∈ OH0 , we may conclude
ArtO ([p])(α) ≡ αq (mod P) ,proving the claim.
Finally we note that by Proposition 2.18 and Lemma 3.6, fractional ideals of O coprime tof(O) ∩ m factor into powers of prime ideals coprime to f(O) ∩ m, so the map ArtO is uniquelydetermined by (7.4).
□
8. Computations of ray class groups of orders
To illustrate how to compute with the exact sequence for change of order and modulus,
we calculate several ray class groups of orders. Let O be an order in a number field K, andlet ρ1 , . . . , ρr be the real embeddings of K. Consider a ray class modulus (m, Σ) for an orderO, where m is an integral O-ideal and Σ = {ρk1 , . . . , ρkℓ }. In the following examples, wewill abbreviate the pair (m, Σ) by the formal product m∞k1 · · · ∞kℓ . We will also denoteprincipal ideals such as αO by (α). We consider a second K-order O′ with O ⊆ O′ and acorresponding modulus (m′ , Σ′ ), requiring m ⊆ m′ and Σ′ ⊆ Σ. We apply Theorem 6.5 with
L = (O; m, Σ) and L′ = (O′ ; m′ , Σ′ ), and make the choice d = (m : O′ ), to obtain the exact
sequence
1→Um′ ,Σ′ (O′ )Um′ (O′ / (m : O′ ))′→× {±1}|Σ\Σ | → Clm,Σ (O) → Clm′ ,Σ′ (O′ ) → 1.
′
Um,Σ (O)Um (O/ (m : O ))(8.1)

The following
series of examples treats several different orders in the real quadratic field√
K = Q( 2) and the computation of some of their ray class groups at moduli ramified
at prime√ ideals lying over (7) √and ∞√of Z. We√let ∞1 ,√∞2 denote the real embeddings
ρi : Q( 2) → R defined by ρ1 ( 2) = 2 and ρ2 ( 2) = − 2, respectively.
The first three examples taken together give information on the corresponding ray classfields. They show that a ray class field of an order can be strictly larger than the compositum

42

GENE S. KOPP AND JEFFREY C. LAGARIAS

of the corresponding ray class field of the maximal order with the ring class field. Using
Theorem 1.3, the class number calculations imply
√
Z[ 2]

√
Z[2 2]√
Z[2 2]

H(7)∞2 · H(1)⊊ H(7)∞2 ,
h
i
h
i
h
i
h
i
√
√
√
√
√
Z[ 2]
Z[2 2]Z[ 2]Z[2 2]
Z[2 2]
since H(7)∞2 H(1): K ≤ H(7)∞2 : K · H(1): K = 6 while H(7)∞2 : K = 12. The
final example presents a calculation in a case where the corresponding ring class field isnontrivial.
√
√
√


8.1. Example 1. Take level data L = Z[ 2]; 7Z[ 2], {ρ2 } = Z[ 2]; (7)∞2 and L′ =
√
√
√


Z[ 2]; Z[ 2], ∅ = Z[ 2]; (1) . Then by (8.1),
√
×
 √ 
 √
 √ 
Z[ 2]×√  → Z[ 2]/(7) × {±1} → Cl(7)∞2 Z[ 2] → Cl Z[ 2] → 1.
1→
U(7)∞2 Z[ 2]√ √ The class group Cl Z[ 2] = 1. Thus, the ray class group Cl(7)∞2 Z[ 2] is isomorphic
to the previous term in the sequence modulo the image of global units. By the Chinese
Remainder Theorem, √×  √√ ×  √√ ×∼Z[ 2]/(7) = Z[ 2]/(3 + 2) × Z[ 2]/(3 − 2)∼
= Z/6Z × Z/6Z.
Moreover,Thus, we see that√√⟨−1, 1 + 2⟩ ∼Z[ 2]×√√ == Z/6Z × Z/2Z.
U(7)∞2 Z[ 2]⟨(1 + 2)6 ⟩ √ Cl(7)∞2 Z[ 2] ∼= Z/6Z.

√√√
8.2. Example 2. Take L = Z[2 2]; 7Z[2 2], {ρ2 } = Z[2 2]; (7)∞2 and, as above, take
√
√
√


L′ = Z[ 2]; Z[ 2], ∅ = Z[ 2]; (1) . Then (8.1) gives √ ×
Z[ 2]
√ ×
 √ 
 √ 
(14)
Z[ 2]

 × {±1} → Cl(7)∞2 Z[2 2] → Cl Z[ 2] → 1.
√ →
1→
√
Z[2 2]
U(7)∞2 Z[2 2]√U(7)(14Z+14 2Z)√ √ As in the previous example, Cl Z[ 2] = 1, so the ray class group Cl(7)∞2 Z[2 2] isisomorphic to the previous term in the sequence modulo the image of global units. By the
Chinese Remainder Theorem, we have √×  √√ ×  √√ ×  √√ 2 ×
Z[
2]/(3
+
2)
×
Z[
2]/(3
−
2)
×
Z[
2]/(
2)
Z[ 2]/(14) ∼=∼= Z/6Z × Z/6Z × Z/2Z.
We also have
and


√
√
U(7) Z[2 2]/(14Z + 14 2Z) = 1
√√Z[ 2]×⟨−1, 1 + 2⟩ ∼√ =√
= Z/6Z × Z/2Z.
U(7)∞2 Z[2 2]⟨(1 + 2)6 ⟩RAY CLASSES FOR ORDERS

We see that

43

 √ 
Cl(7)∞2 Z[2 2] = 12.

With more careful accounting, we can obtain an isomorphism √ Cl(7)∞2 Z[2 2] ∼= Z/6Z × Z/2Z.
√ 
8.3. Example 3. We compute the same group Cl(7)∞2 Z[2 2] a different way. Take L =√√Z[2 2]; (7)∞2 as above, but instead take L′ = Z[2 2]; (1) . Then (8.1) gives
√
 √
×
 √ 
 √ 
Z[2 2]×√  → Z[2 2]/(7) × {±1} → Cl(7)∞2 Z[2 2] → Cl Z[2 2] → 1.
1→
U(7)∞2 Z[2 2]√ 
The ring class group Cl Z[2 2] = 1. By the Chinese Remainder Theorem,×  √ √√ ×  √√ ×∼Z[2 2]/(7) = Z[2 2]/(1 + 2 2) × Z[2 2]/(1 − 2 2)∼
= Z/6Z × Z/6Z.
The quotient of global unit groups is√√Z[2 2]×⟨−1, (1 + 2)2 ⟩ ∼√√ == Z/2Z × Z/3Z ∼= Z/6Z.
U(7)∞2 Z[2 2]⟨(1 + 2)6 ⟩It follows that √ Cl(7)∞2 Z[2 2] ∼= Z/6Z × Z/2Z.

√√

8.4. Example 4. Take L = Z[5 2]; (7)∞2 and L′ = Z[5 2]; (1) . Then (8.1) gives
√
×
 √ 
 √ 
 √
Z[5 2]×√  → Z[5 2]/(7) × {±1} → Cl(7)∞2 Z[5 2] → Cl Z[5 2] → 1.
1→
U(7)∞2 Z[5 2]In a similar method to the above examples (e.g., by another change of order calculation), we
can compute
 √ 
 √ 
Cl Z[5 2] := Cl(1) Z[5 2] ∼= Z/2Z.
√
The ring class number is 2. As a consequence, by Theorem 1.3, the ring class field for Z[5 2]is a quadratic extension of K, so it is a degree 4 extension of Q. We also have× √Z[5 2]/(7) × {±1} ∼= Z/6Z × Z/6Z × Z/2Z;√ ×√ 3
Z[5 2]⟨−1, (1 + 2) ⟩ ∼√ =√= Z/2Z × Z/2Z.
U(7)∞2 Z[5 2]⟨(1 + 2)6 ⟩We see that √ (6 · 6 · 2)(2)
Cl(7)∞2 Z[5 2] =
= 36.
2·2
To determine the ray class group structure, it would be necessary to compute the maps in
the exact sequence.

44

GENE S. KOPP AND JEFFREY C. LAGARIAS

9. Concluding remarks
The main results of this paper assigned ray class fields to invertible ray class groups oforders defined for level data specifying an order and a modulus (an ideal and a set of realplaces). The constructed class fields were characterized in terms of splitting conditions on
their prime ideals.
There are several other aspects of class field theory, specified by the main theorems ofclass field theory listed by Hasse [28], that might have analogues in the ray class field theoryof orders.
(1) Norms of ideals in orders. The Takagi class field theory has an interpretation of
ray class groups in which the kernels of some group maps involve groups generated
by norms of ideals. There is a natural way to define norms of integral ideals a inorders, as NmO (a) = |O/a|. Norms of fractional ideals are discussed in Appendix A.
One subtlety that arises is that norms of non-invertible ideals are in general not
multiplicative.
(2) Zeta functions of orders. Zeta functions played an important role in the historicaldevelopment of class field theory. We can associate zeta functions and L-functions toorders. There is a zeta function whose terms involve norms of invertible integral idealsof orders. There is another zeta function obtained by summing over norms of a larger
set of ideals of an order, including non-invertible ideals. The uniqueness of primary
decomposition (Proposition 2.2) implies that both of these zeta functions have Euler
products; the latter one will have unusual factors at the finite set of maximal idealsof O that contain the conductor ideal f(O).
(3) Ray class monoids and L-functions. If we allow non-invertible ideals, then the rayclass group is enlarged to become a ray class monoid. We treat aspects of the structure of ray class monoids of orders in [33]. Additionally one can define L-functions
associated to characters of ray class groups of orders. If one allows non-invertible
ideals, then one can also allow new L-functions using characters of ray class monoids;see [12, 13, 29, 41–43].
There is a development of class field theory for orders inside an idèlic framework formulated by Pengo [47] and detailed by Campagna and Pengo [10, Sec. 4] (see in particular
their Defn. 4.1). It remains to relate that idèlic definition to the ray class group defined in
Definition 7.1 of this paper. The equivalence of the two notions of class fields of orders (inthe case of no ramification at infinity considered by Campagna and Pengo) has not been
definitively established (but see Remark 6.8).
Acknowledgments. The authors thank Pete L. Clark, Francesco Campagna and Riccardo
Pengo, and John Voight for helpful comments and references. The first author was partially
supported by the NSF grant DMS-2302514 and by University of Bristol, the Heilbronn
Institute for Mathematical Research, and Purdue University. He thanks Trevor Wooley
for support and for helpful mathematical conversations. The second author was partially
supported by NSF grant DMS-1701576.
Appendix A. Norms of ideals in orders of number fields
Let K be a number field and O an order in K. We give a criterion for multiplicativity of(absolute) norms of integral ideals of an order to hold; it does not hold in general. We use

RAY CLASSES FOR ORDERS

45

this criterion to extend the notion of norm of an integral ideal of O to norm of a fractional
ideal of O. We discuss the effect of change of order on norms of fractional ideals.
Definition A.1. Let a be an integral O-ideal. If a is nonzero, define the norm of a to be
NmO (a) = [O : a], where [O : a] = |O/a| is the index of a in O as an abelian group. Define
NmO (0) = 0.
For invertible ideals a, b, the norm is multiplicative: Nm(ab) = NmO (a) NmO (b). The
following proposition shows the stronger result that the norm of the product of two integral
ideals is multiplicative whenever one of them is invertible.
Proposition A.2. Let a ∈ I∗ (O) and b ∈ I(O) (so a is invertible, whereas b may or may
not be invertible). Then, NmO (ab) = NmO (a) NmO (b).
Proof. If b = 0, then NmO (ab) = 0 = NmO (a) NmO (b). Assume from now on that b ̸= 0.
The norm of ab and the norm of a are related by the following equation:
NmO (ab) = [O : ab] = [O : a][a : ab] = NmO (a)[a : ab].

(A.1)

Let {p1 , . . . , pk } be the set of maximal ideals of O containing ab. Using primary decomposition (Proposition 2.2), we may write
a=

k
\
j=1

qj =

k
Y

qj ,

b=

j=1

k
\
j=1

rj =

k
Y

rj ,

j=1

where qj and rj are separately either primary ideals having radical pj , or else equal to theunit ideal O. Locally, aOpj = qj Opj and bOpj = rj Opj . Moreover since a is invertible, by
Proposition 5.8 it is locally principal, so we may write aOpj = αj Opj for 1 ≤ j ≤ k. Choosesome α ∈ O such that α ≡ αj (mod qj rj ) for 1 ≤ j ≤ k. These conditions imply α ∈ a,
Define an additive group homomorphism (indeed, an isomorphism of O-modules)
ϕ : O → a/abby ϕ(x) = αx + ab.
We first show that ϕ is surjective. Consider y ∈ a. Locally in Opj , write y = αj xj forsome xj ∈ Opj . Choose some x ∈ O such that x ≡ xj (mod qj rj ) for 1 ≤ j ≤ k. Thus,y ≡ αx (mod qj rj ) for 1 ≤ j ≤ k, soy − αx ∈k\qj rj =

j=1

kY

qj rj = ab.

j=1

That is, ϕ(x) = y + ab.
We now compute the kernel of ϕ. We have ϕ(x) = 0 if and only if αx ∈ ab. Now
α ∈ a, so αx ∈ ab whenever x ∈ b. Conversely, suppose αx ∈ ab. Then, in the local
ring Opj , we have αx ∈ abOpj = αj rj Opj . Also, α − αj ∈ qj rj Opj = αj rj Opj , and thus
αj x = αx − (α − αj )x ∈ αj rj Opj . Dividing, x ∈ rj Opj , so x ∈ rj (because x ∈ O). Thus,
x∈

k
\
j=1

So ker ϕ = b.

rj = b.

46

GENE S. KOPP AND JEFFREY C. LAGARIAS

By the first isomorphism theorem, there is an isomorphism of abelian groups (indeed, of
O-modules)
O/b ∼= a/ab.
Equating the sizes of the two abelian groups, [a : ab] = [O : b] = NmO (b). Substituting into(A.1), NmO (ab) = NmO (a) NmO (b).
□The norm also enjoys a multiplicativity property for coprime ideals, allowing the norm to
be calculated from its values on primary ideals.
Proposition A.3. If a, b are coprime integral O-ideals, then NmO (ab) = NmO (a) NmO (b).
In particular, if m is any nonzero integral O-ideal with primary decomposition
n
n
\
Y
m=
qi =
qi ,
i=1

then its norm is given by NmO (m) =

i=1

Qn

i=1 NmO (qi ).

Proof. When a or b is zero, NmO (ab) = 0 = NmO (a) NmO (b). Otherwise, ab = a ∩ b, andby the Chinese Remainder Theorem [24, Ch. 7, Thm. 17],O/(ab) = O/(a ∩ b) ∼= O/a ⊕ O/b,so NmO (ab) = |O/(ab)| = |O/a| · |O/b| = NmO (a) NmO (b).
Primary ideals associated to different maximal ideals are coprime, so the norm of annonzero integral O-ideal is the product of the norms of its primary constituents.
□
The norm need not be multiplicative if both ideals a, b are non-invertible and non-coprime.
Marseglia [40] observed in a single order O instances of both strict submultiplicativity
NmO (ab) < NmO (a) NmO (b) and strict supermultiplicativity NmO (ab) > NmO (a) NmO (b).
Example A.4 (Super-multiplicativity and sub-multiplicativity of norms). The multiplicativity condition NmO (ab) = NmO (a) NmO (b) for a, b ∈ I(O) can only fail when both a andb are not invertible. In particular, neither a nor b can be coprime to the conductor f(O).
(1) An example of strict super-multiplicativity of norms was given in Example 2.14. It
showed in O = Z[2i] having conductor ideal Q2 = 2OK , where OK = Z[i], that:
8 = NmO ((Q2 )2 ) > (NmO (Q2 ))2 = 4.
(2) An example of strict sub-multiplicativity of norms are given by Marseglia [40, Ex. 3.4].
The example takes K = Q(α), where α is a root of a monic irreducible polynomialof degree 4 with integer coefficients, such as x4 − x − 1. Consider the order
O := Z + pZ[α] = Z + pαZ + pα2 Z + pα3 Z,
where p ≥ 5 is a rational prime number. Then the lattices
a := pZ + pαZ + p2 α2 Z + p2 α3 Z andb := pZ + p2 αZ + pα2 Z + p2 α3 Z.
are O-ideals, andab = p2 Z + p2 αZ + p2 α2 Z + p2 α3 Z.
We havep5 = NmO (ab) < NmO (a) NmO (b) = p3 · p3 = p6 .

RAY CLASSES FOR ORDERS

47

Marseglia gave an example of strict super-multiplicativity in this order O as well.
Proposition A.2 will justify an extension of the norm to fractional ideals.
Definition A.5. Let d ∈ J(O), and write d = (a : b) = ab−1 for some a ∈ I(O) andNmO (a)b ∈ I∗ (O). Define NmO (d) = Nm
. (The next proposition shows this norm is well-defined.)
O (b)Proposition A.6. The norm of a fractional ideal d in Definition A.5 is well-defined. If
d ⊆ O, then its fractional ideal norm agrees with its integral ideal norm. If c ∈ J∗ (O) andd ∈ J(O), then NmO (cd) = NmO (c) NmO (d).
Proof. If d ∈ J(O) and d = a1 b−1= a2 b−1
for some a1 , a2 ∈ I(O) and b1 , b2 ∈ I∗ (O),12then a1 b2 = a2 b1 , so NmO (a1 ) NmO (b2 ) = NmO (a2 ) NmO (b1 ) by Proposition A.2. Thus,
NmO (a1 )
NmO (a2 )= Nm, so Nm(d) is well-defined.
NmO (b1 )O (b2 )In the case d ⊆ O, we may choose a1 = d and b1 = O, so the integral ideal norm agrees
with the fractional ideal norm.
−1
Now, consider c ∈ J∗ (O) and d ∈ J(O). Write c = a1 b−1
1 and d = a2 b2 for a1 , b1 , b2 ∈∗−1
J (O) and a2 ∈ J(O). Then, cd = (a1 a2 )(b1 b2 ) , soNmO (a1 ) NmO (a2 )NmO (a1 a2 )== NmO (c) NmO (d),NmO (b1 b2 )NmO (b1 ) NmO (b2 )using Proposition A.2 and Definition A.5.
NmO (cd) =□Example A.7 (Behavior of norms of non-integral fractional ideals having an ideal powerthat is an integral ideal). Consider the non-maximal order O = Z[2i] of the Gaussian field
K = Q(i), with maximal order OK = Z[i].
(1) The (non-integral) fractional O-ideal
r1 := (1 + i)O = 4Z + (1 + i)Z,is a principal fractional O-ideal, hence it is an invertible fractional O-ideal. Herer1 = cd−1 where c = 2(1 + i)O and d = 2O are invertible integral O-ideals, fromwhich we may compute NmO (r1 ) = 2.
Recall from Example 3.10 that r1 has ideal square r21 = 2iO = q′4 , which wasshown to be an irreducible integral O-ideal in Example 2.14, having NmO (q′4 ) = 4.
The fractional O-ideal r1 then has fractional O-ideal norm NmO (r1 ) = 2.
The equality NmO (r21 ) = (NmO (r1 ))2 = 4 is consistent with Proposition A.6, since
r1 is an invertible ideal.
(2) The (non-integral) fractional O-ideal
r2 := (1 + i)OK = 2Z + (1 + i)Z,is a non-invertible fractional O-ideal. Viewed as a fractional O-ideal, NmO (r2 ) = 2,
using r2 = c2 (d2 )−1 with c2 = 2(1 + i)OK and d2 = 2O. Next we have(r2 )2 = 2OK = Q2 ,and r22 is a non-invertible O-ideal since it is an OK -ideal. We have NmO (Q2 ) = 2.
Consequently,
2 = NmO (r22 ) ̸= (NmO (r2 ))2 = 4,
showing that the hypothesis of O-invertibility of at least one of c and d cannot berelaxed in Proposition A.6.

48

GENE S. KOPP AND JEFFREY C. LAGARIAS(3) The (non-integral) fractional O-ideal
r3 := iO = 2Z + iZ
is an invertible fractional O-ideal and it has square (r3 )2 = O, which is an invertible
integral O-ideal. In this case, NmO (r3 ) = 1.
The final proposition shows that norms of invertible fractional O-ideals are preserved
under extension.
Proposition A.8. Suppose O ⊆ O′ for K-orders, and let ext be the extension map fromfractional ideals on O to fractional ideals on O′ . If c ∈ J∗ (O), then NmO′ (ext(c)) = NmO (c).
Proof. By the multiplicativity property of the norm proven in Proposition A.6 and the factthat extension preserves invertibility, it suffices to prove the claim for integral ideals; we canmultiply c by a positive integer so it is contained in O, so without loss of generality, assume
c is integral.
By definition ext(c) = cO′ , and we wish to show
NmO′ (cO′ ) = NmO (c).
Firstly, since cO′ is an integral O′ ideal, we have
NmO′ (cO′ ) = [O′ : cO′ ].
We also have, purely as Z-modules, the index relations[O′ : cO′ ] = [O′ : O][O : cO′ ] = [O′ : O] NmO (cO′ ).
Since c is an invertible O-ideal, we have
NmO (cO′ ) = NmO (c) NmO (O′ ) by Proposition A.6.
Combining these equalities yields
NmO′ (cO′ ) = NmO (c) (NmO (O′ )[O′ : O]) .
Consequently the lemma for c is equivalent to the assertion that
NmO (O′ ) = [O′ : O]−1 .
To show this, pick a positive integer d ∈ N so that dO′ ⊆ O; the choice d = [O′ : O] works.
Then NmO (dO′ ) = [O : dO′ ]. Consider dO′ = (dO)(O′ ) as a product of O-ideals, where dO
is an invertible O-ideal, Proposition A.6 applies to giveNmO (dO′ ) = NmO (dO) NmO (O′ ).
We suppose [K : Q] = n, whence NmO (dO) = [O : dO] = dn , andNmO (dO′ ) = dn NmO (O′ ).

(A.2)

Next, since dO′ ⊆ O ⊆ O′ , a Z-module index calculation gives
NmO (dO′ ) = [O : dO′ ] =[O′ : dO′ ]= dn [O′ : O]−1 .
[O′ : O]

Comparison with (A.2) gives NmO (O′ ) = [O′ : O]−1 , as required.

□RAY CLASSES FOR ORDERS

49

Example A.9 (Change of norm under extension for non-invertible ideals). The norm ofa non-invertible O-ideal may change under extension. For any non-maximal order O the(absolute) conductor ideal f(O) = fOK (O) is non-invertible integral O-ideal that is an OK ideal, whose norm will always increase under extension to OK . We have ext(f(O)) = f(O),and NmO (f(O)) = [O : f(O)], while
NmOK (f(O)) = [OK : f(O)] = [OK : O][O : f(O)] = [OK : O] NmO (f(O)).
The usefulness of Proposition A.8 is that it applies to invertible fractional ideals c that are
not coprime to the relative conductor ideal fO′ (O). For the non-maximal order O = Z[2i] of
the Gaussian field K = Q(i), treated in Example A.7, the ideal r1 := (1+i)O = 4Z+(1+i)Z
is a principal fractional O-ideal so is O-invertible. It is not coprime to the conductor idealf(O) = fOK (O) = 2OK . Noting that r1 OK = (1 + i)OK , it follows that NmOK (r1 OK ) =NmO (r1 ) = 2, consistent with Proposition A.8.
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Department of Mathematics, Louisiana State University, Baton Rouge, LA, USA
Email address: kopp@math.lsu.edu
Department of Mathematics, University of Michigan, Ann Arbor, MI, USA
Email address: lagarias@umich.edu