Math. J. Okayama Univ. 65 (2023), 117–123

A NOTE ON FIELDS GENERATED BY JACOBI SUMS
Yuichiro HoshiAbstract. In the present paper, we study fields generated by Jacobisums. In particular, we completely determine the field obtained byadjoining, to the field of rational numbers, all of the Jacobi sums “of twovariables” with respect to a fixed maximal ideal of the ring of integers
of a fixed prime-power cyclotomic field.

Introduction
Throughout the present paper, let us fix• a prime number l,• a positive integer N , and• a maximal ideal p of the ring o of integers of the finite Galois extension K of Q obtained by adjoining to Q a primitive lN -th root ofunity.
defWrite G = Gal(K/Q) for the Galois group of the finite Galois extension
def

K/Q, D ⊆ G for the decomposition subgroup associated to p, κ(p) = o/p
for the residue field at p, and p for the characteristic of κ(p). Suppose thatp 6= l. Write, moreover, χ : κ(p)× → µlN (K) ⊆ K × for the homomorphism
determined by the lN -th power residue symbol at p. Following [6], for eachpositive integer n and each element a = (a1 , . . . , an ) of Zn , let us define theJacobi sum associated to a ∈ Zn as follows [cf. [6], (I)]:defn+1

ja = (−1)

XnY

χ(xi )ai ∈ o.

× n i=1(x1 ,...,xP n )∈(κ(p) )x=−1j jIn the present paper, we discuss intermediate extensions of the finite Galois
extension K/Q obtained by adjoining to Q Jacobi sums. Let us recall that T.
Ono, M. Kida, and A. Gyoja [cf. [2], [3], [5]] and N. Aoki [cf. [1]] have studied
these intermediate extensions. Note that we have an equality Q(jb ; b ∈Z2 ) = Q(jc ; c ∈ Zm , m ≥ 1) [cf. Proposition 2.1, (ii)].
The main result of the present paper is as follows:
Theorem A. The following assertions hold:(i) If K D is totally real, then Q(ja ; a ∈ Z2 ) = Q.
Mathematics Subject Classification. 11L05, 11R18.
Key words and phrases. Jacobi sum.
117

118

Y. HOSHI(ii) If K D is not totally real, then Q(ja ; a ∈ Z2 ) = K D .
Note that one verifies easily that Theorem A in the case where l is odd, andN = 1 [i.e., in the “odd prime cyclotomic field case”] may also be derived
from [2], Theorem 2, together with a similar argument to the argument
applied in the proof of Theorem A. The author of the present paper will
apply Theorem A to the study of geometrically pro-l anabelian geometry
for tripods over finite fields. Then this “known” result [i.e., Theorem A in
the “odd prime cyclotomic field case”] is not sufficient for this application.
Moreover, Theorem A could not be found in literature. This is one main
motivation of the study of the “prime-power cyclotomic field case” in thepresent paper.
1. Some Lemmas
def

We shall write Λ = Z/lN Z. Moreover, for each t ∈ Λ× , we shall write σt ∈G for the unique element which induces the t-th power map on the subgroupµlN (K) ⊆ K × . Then one verifies easily that the assignment “t 7→ σt ”
∼determines an isomorphism Λ× → G of groups. Moreover, one also verifies
easily that the subgroup D ⊆ G coincides with the subgroup hσp i ⊆ G∼generated by σp , i.e., corresponds, via the isomorphism Λ× → G, to thesubgroup of Λ× generated by the image of p in Λ× .
Definition 1.1. Let m be a positive integer. Then we shall write L[m] ⊆ Z
for the set of integers et such that the inequalities 0 < et < m hold, and,emoreover, the integer t is prime to m.

Remark. Observe that one verifies easily that, in the situation of Definition 1.1, the natural surjective map Z ։ Z/mZ restricts to a bijective map∼
L[m] → (Z/mZ)× .
Lemma 1.2. The following assertions hold:(i) Let m be a positive integer, et an element of L[m], r a positive integer,and d an element of {0, 1}. For a rational number s ∈ Q, write [s]for the “integral part” of s [i.e., the largest integer which is less thandefor equal to s] and hsi = s − [s] for the “fractional part” of s. ThenhDet Ei(rm + d)= ret.
m(ii) Let N0 be a positive integer such that N0 ≤ N . For each positivet such thatinteger r, write r + L[lN0 ] ⊆ Z for the set of integers e

A NOTE ON FIELDS GENERATED BY JACOBI SUMS

et − r ∈ L[lN0 ]. Then
L[lN ] =lN−N
G0 −1

119

(ilN0 + L[lN0 ]).

i=0Proof. These assertions are immediate.

Lemma 1.3. Let ρ : Λ× → C× be an odd Dirichlet character. ThenXet · ρ(t) 6= 0et∈L[lN ]— where, for each et ∈ L[lN ], we write t ∈ Λ× for the image of et in Λ [cf.
Remark following Definition 1.1].
Proof. Writedef
N0 = min i ∈ {1, 2, . . . , N } | 1 + li Λ ⊆ Ker(ρ)
defand Λ0 = Z/lN0 Z. Then one verifies easily that the odd, hence also nontrivial, Dirichlet character ρ : Λ× → C× factors as the composite of the naturalsurjective homomorphism Λ× ։ Λ×
0 and a primitive odd Dirichlet character××ρ0 : Λ0 → C . Now let us observe that since ρ0 is odd, hence also nontrivial,
it follows that, for each integer i such that 0 ≤ i ≤ lN −N0 − 1,XXXXeet · ρ0 (t) =t · ρ0 (t)ρ0 (t) +t) · ρ(t) = ilN0 ·(ilN0 + eet∈L[lN0 ]t∈Λ×0et∈L[lN0 ]et∈L[lN0 ]e— where, for each et ∈ L[lN0 ], we write t ∈ Λ×0 for the image of t in Λ0[cf. Remark following Definition 1.1]. Thus, it follows immediately from
Lemma 1.2, (ii), thatX
Xeet · ρ(t) = lN −N0 ·t · ρ0 (t).
et∈L[lN ]et∈L[lN0 ]In particular, to verify Lemma 1.3, we may assume without loss of generality,by replacing “(N, ρ)” by (N0 , ρ0 ), that ρ0 is primitive. On the other hand,if ρ0 is primitive, then Lemma 1.3 is well-known [cf., e.g., [4], Chapter VII,
§2, Exercise 4]. This completes the proof of Lemma 1.3.
Lemma 1.4. The following assertions hold:(i) Let T be a group of order 2, C a cyclic 2-group, and H a subgroup ofT × C that does not contain the subgroup T × {1}. Then there exist
not necessarily distinct two subgroups H1 , H2 of T × C such that

120Y. HOSHI• T × {1} 6⊆ H1 , T × {1} 6⊆ H2• both (T × C)/H1 and (T × C)/H2 are cyclic, and, moreover,• H = H1 ∩ H2 .
(ii) Suppose that K D is not totally real. Then there exist not necessarily
distinct two odd Dirichlet characters ρ1 , ρ2 : Λ× → C× such thatthe intersection Ker(ρ1 ) ∩ Ker(ρ2 ) coincides with the subgroup of Λ×
generated by the image of p in Λ× .
Proof. First, we verify assertion (i). If the quotient of T × C by H is cyclic,
defdefthen the subgroups H1 = H, H2 = H satisfy the desired condition in thestatement of assertion (i). Thus, to verify assertion (i), we may assume
without loss of generality that the quotient of T × C by H is not cyclic,
which implies that there exists an element c0 of the group C of order 2. In
particular, to verify assertion (i), we may assume without loss of generality— by replacing “T ”, “C” by the respective images of T , C in (T × C)/H —that H = {1}. Then if one writes t0 ∈ T for the unique nontrivial elementof T and H1 ⊆ T × C for the subgroup of T × C generated by (t0 , c0 ),defthen one verifies easily that the subgroups H1 , H2 = {1} × C satisfy thedesired condition in the statement of assertion (i). This completes the proofof assertion (i).
Next, we verify assertion (ii). If l is odd, then since Λ× is cyclic, assertion (ii) is immediate. If l = 2, then assertion (ii) follows immediatelyfrom assertion (i). This completes the proof of assertion (ii), hence also ofLemma 1.4.

2. Proof
It seems to the author that the three assertions discussed in Proposition 2.1 below are likely to be well-known. However, the author decided togive proofs of these assertions here for the sake of the reader and the sakeof completeness.
Proposition 2.1. Let n be a positive integer and a = (a1 , . . . , an ) an element of Zn . Then the following assertions hold:(i) The inclusion ja ∈ K D holds.
(ii) The inclusion ja ∈ Q(jb ; b ∈ Z2 ) holds.
(iii) Suppose that K D is totally real. Then the inclusion ja ∈ Q holds.
Proof. First, we verify assertion (i). Let us first observe that it is immediatethat the homomorphism χ : κ(p)× → µlN (K) ⊆ K × is D-equivariant, i.e.,
relative to the respective natural actions of D on κ(p)× and K × . Thus,assertion (i) follows immediately from the definition of the Jacobi sum ja .
This completes the proof of assertion (i).

A NOTE ON FIELDS GENERATED BY JACOBI SUMS

121

Next, we verify assertion (ii). Let us recall from the first and second
displays of [6], p.492, that if n ≥ 3, and a1 +a2 ∈ lN Z (respectively, a1 +a2 6∈
lN Z), thenja = j(a2 ) · j(a3 ,...,an ) · #κ(p)(respectively, ja = j(a1 +a2 ) · j(a1 ,a2 ) · j(a1 +a2 ,a3 ,...,an ) ).
Thus, assertion (ii) follows immediately from the easily verified fact that theJacobi sum in the case where n = 1 is contained in {±1}. This completes
the proof of assertion (ii).
Finally, we verify assertion (iii). Let us first observe that one verifies
easily that either lN = 2 or #D ∈ 2Z. If lN = 2, which implies that Q = K,then assertion (iii) is immediate. Suppose that #D ∈ 2Z, which implies
that #κ(p) is the square of a rational number. If {a1 , . . . , an } ⊆ lN Z, thenit follows from the equalityn −1
ja = #κ(p) · 1 − 1 − #κ(p)of [6], (2), that assertion (iii) holds. Suppose that {a1 , . . . , an } 6⊆ lN Z.
Then since #κ(p) is the square of a rational number as mentioned above,and ja ∈ K D [cf. assertion (i)], which implies that ja is a real number, it
follows from the equality|ja |2 = #κ(p)s−2
defdef P— where we write an+1 = ni=1 ai and s = #{ i ∈ {1, . . . , n+1} | ai ∈ lN Z }— of [6], (10), that assertion (iii) holds. This completes the proof of assertion(iii), hence also of Proposition 2.1.
Definition 2.2. Let n be a positive integer and a an element of Zn . Then weshall write Q(ja ) for the intermediate extension of the finite Galois extension
K/Q that corresponds to the subgroup of G consisting of the elements whoseactions on o preserve the principal ideal of o generated by ja ∈ o.
Remark. Observe that it follows immediately from Proposition 2.1, (i),(ii), together with the various definitions involved, that, in the situation ofDefinition 2.2, we have inclusions
Q ⊆ Q(ja ) ⊆ Q(ja ) ⊆ Q(jb ; b ∈ Z2 ) ⊆ K D ⊆ K.
Definition 2.3. Let n be a positive integer. Then we shall write (1[n] ) ∈ Zn
for the element of Zn each of whose n components is given by 1 ∈ Z.
Lemma 2.4. Suppose that K D is not totally real. Let r be a positive integer
and d an element of {0, 1}. Then Q(j(1[rlN +d] ) ) = Q(j(1[rlN +d] ) ) = K D .

122

Y. HOSHI

Proof. Let us first observe that it follows from the displayed inclusions ofRemark following Definition 2.2 that, to verify Lemma 2.4, it suffices toverify the inclusion K D ⊆ Q(j(1[rlN +d] ) ). Next, let us also observe that itfollows immediately from [6], (8), that the assignment “p 7→ j(1[rlN +d] ) ” is afunction of type (S) in the sense of [5], §1, i.e., in the case where we take the“(k, K)” of [5], §1, to be (Q, K); moreover, it follows from [6], (9), togetherwith Lemma 1.2, (i), that the “ω” of [5], §1, for this function of type (S) isgiven byX−1
ret · σ−tet∈L[lN ]— where, for each et ∈ L[lN ], we write t ∈ Λ× for the image of et in Λ[cf. Remark following Definition 1.1]. In particular, it follows from [5],(2.4), that, for each σ ∈ G, this element σ is contained in the subgroup∼
Gal(K/Q(j(1[rlN +d] ) )) ⊆ G if and only if this element σ ∈ G → Λ× is contained in the kernel of every Dirichlet character ρ : Λ× → C× such that ρmaps the image of p in Λ× to 1 ∈ C× , and, moreover,X
et · ρ(t) 6= 0.
et∈L[lN ]Thus, the desired inclusion K D ⊆ Q(j(1[rlN +d] ) ) follows immediately fromLemma 1.3 and Lemma 1.4, (ii). This completes the proof of Lemma 2.4. Remark. Note that [2], Theorem 2, may be regarded as Lemma 2.4 in thecase where l is odd, and the equality (N, r, d) = (1, 1, 1) holds. Moreover,Lemma 2.4 in the case where l is odd, and the equality (N, r, d) = (1, 1, 1)
holds may also be derived from [1], Lemma 6.2 [cf. also [1], Remark 6.6].
On the other hand, no result that claims explicitly the two equalities in thestatement of Lemma 2.4 in the case where N > 1 [i.e., the “prime-powercyclotomic field case”] could be found in literature. Here, let us recall that
one may find [1], Theorem 0.4, that establishes a concrete description of thefield Q(ja ) in the “prime-power cyclotomic field case” [cf. also some results
proved in [1], §7].
Proof of Theorem A. Assertion (i) follows from Proposition 2.1, (iii). Next,
we verify assertion (ii). Let us first observe that it follows from Lemma 2.4that Q(j(1[lN ] ) ) = K D . Thus, assertion (ii) follows from the displayed inclusions of Remark following Definition 2.2. This completes the proof ofTheorem A.
A NOTE ON FIELDS GENERATED BY JACOBI SUMS

123

Acknowledgements
The author would like to thank A. Tamagawa for explaining an argument
concerning the topic discussed in the present paper. This discussion is one
main motivation of the study of the present paper. The author also would
like to thank the referee for some helpful comments. This research was supported by JSPS KAKENHI Grant Number 21K03162 and by the ResearchInstitute for Mathematical Sciences, an International Joint Usage/ResearchCenter located in Kyoto University.
References[1] N. Aoki: Abelian fields generated by a Jacobi sum. Comment. Math. Univ. St. Paul.
45 (1996), no. 1, 1–21.
[2] A. Gyoja and T. Ono: A note on Jacobi sums. II. Proc. Japan Acad. Ser. A Math. Sci.
69 (1993), no. 4, 91–93.
[3] M. Kida and T. Ono: A note on Jacobi sums. Proc. Japan Acad. Ser. A Math. Sci. 69(1993), no. 2, 32–34.
[4] J. Neukirch: Algebraic number theory. Grundlehren der Mathematischen Wissenschaften, 322. Springer-Verlag, Berlin, 1999.
[5] T. Ono: A note on Jacobi sums. III. Proc. Japan Acad. Ser. A Math. Sci. 69 (1993),
no. 7, 272–274.
[6] A. Weil: Jacobi sums as “Grössencharaktere”. Trans. Amer. Math. Soc. 73, (1952).
487–495.
Yuichiro Hoshi
Research Institute for Mathematical Sciences
Kyoto University
Kyoto, 606-8502, Japane-mail address: yuichiro@kurims.kyoto-u.ac.jp(Received November 24, 2021 )(Accepted March 29, 2022 )