arXiv:0704.1557v1  [math.RA]  12 Apr 2007
On the residue fields of Henselian valued stable fields, II
I.D. Chipchakov ∗
Introduction
This paper is a continuation of [Ch2]. Let E be a field, Esep a separable closure of
E , E∗ the multiplicative group of E , d(E) the class of finite-dimensional central
division E -algebras, Br (E) the Brauer group of E , Gal (E) the set of finite Galois
extensions of E in Esep , and NG (E) the set of norm groups N(M/E): M ∈ Gal(E) .
Let also P be the set of prime numbers, Br (E)p the p -component of Br (E) , for each
p ∈ P , P(E) the set of those p ∈ P , for which E is properly included in its maximal
p -extension E(p) (in Esep ), and Π(E) the set of all p
′ ∈ P , for which the absolute
Galois group GE = G(Esep/E) is of nonzero cohomological p
′ -dimension cd p′(GE) .
Recall that E is p -quasilocal, for a given p ∈ P , if p 6∈ P(E) or the relative
Brauer group Br (F/E) equals the subgroup pBr(E) = {bp ∈ Br(E): pbp = 0} , for
every cyclic extension F of E of degree p . The field E is said to be primarily
quasilocal (PQL), if it is p -quasilocal, for all p ∈ P ; E is called quasilocal, if its
finite extensions are PQL. We say that E is a strictly PQL-field, if it is PQL and
Br (E)p 6= {0} , for every p ∈ P(E) (see [Ch4, Corollary 3.7], for a characterization
of the SQL-property). The purpose of this note is to complement the main result of
[Ch2] and to shed light on the structure of Br (K) , where K is an absolutely stable
field (in the sense of E. Brussel) with a Henselian valuation whose value group is
totally indivisible.
1. Subfields of central division algebras over PQL-fields
Assume that E is a PQL-field and M ∈ Gal(E) , such that G(M/E) is nilpotent, and
let R be an intermediate field of M/E . It follows from Galois theory, [Ch2, Theorem
4.1 (iii)], the Burnside-Wielandt characterization of nilpotent finite groups, and the
general properties of central division algebras and their Schur indices (see [KM, Ch.
6, Sect. 2] and [P, Sects. 13.4 and 14.4]) that R embeds as an E -subalgebra in each
∗ Partially supported by Grant MI-1503/2005 of the Bulgarian Foundation for Scien-
tific Research.
http://arxiv.org/abs/0704.1557v1
algebra D ∈ d(E) of index divisible by [R: E] . In this Section, we demonstrate the
optimality of this result in the class of strictly PQL-fields.
Theorem 1.1. For each nonnilpotent finite group G , there exists a strictly PQL-
field E = E(G) with Br (E) ∼= Q/Z and Gal (E) containing an element M such that
G(M/E) ∼= G and M does not embed as an E -subalgebra in any ∆ ∈ d(E) of index
equal to [M:E] .
Proof. Let P(G) be the set of prime divisors of the order of G . Our argument relies
on the existence (cf. [Ch1, Sect. 4]) of an algebraic number field E0 , such that there
is M0 ∈ Gal(E0) with G(M0/E0) ∼= G , and E0 possesses a system of (real-valued)
valuations {w(p): p ∈ P(G)} satisfying the following conditions:
(1.1) (i) w(p) extends the normalized p -adic valuation (in the sense of [CF, Ch.
VII]) of the field Q of rational numbers, for each p ∈ P(G) ;
(ii) The completion M0,w(p)′ lies in Gal (E0,w(p)) and G(M0,w(p)′/E0,w(p)) is isomor-
phic to a Sylow p -subgroup of G(M0/E0) , whenever p ∈ P(G) and w(p)′ extends
w(p) on M0 .
Let Ap be the maximal abelian p -extension of E0 in M0 , for each p ∈ P(G) . It
follows from the nonnilpotency of G , the Burnside-Wielandt theorem and Galois
theory that M0/E0 has an intermediate field F 6= E0 such that F ∩ Ap = E , for
all p ∈ P(G) (see [Ch1, Sect. 4]). Fix a prime divisor π of [F: E0] as well as Sylow
π -subgroups Hπ ∈ SylπG(M0/F) and Gπ ∈ SylπG(M0/E0) with Hπ ⊂ Gπ , and
denote by Fπ and Eπ the maximal extensions of E0 in M0 fixed by Hπ and
Gπ , respectively. It is clear from (1.1) (ii) and [CF, Ch. II, Theorem 10.2] that Eπ
has a valuation t(π) extending w(π) so that M0 ⊗Eπ Eπ,t(π) is a field. This means
that t(π) is uniquely (up-to an equivalence) extendable to a valuation µ(π) of M0 ,
and allows us to identify E0,w(π) with Eπ,t(π) . Using repeatedly the Grunwald-Wang
theorem [W] and the normality of maximal subgroups of finite π -groups (cf. [L, Ch. I,
Sect. 6]), one also obtains that there is a finite extension K of E0 in E0(π) , such that
K⊗E0 E0,w(π) is a field isomorphic to Fπ,ν(π) as an E0 -algebra, where ν(π) is the
valuation of Fπ induced by µ(π) . These observations indicate that K ∩M0 = E0 ,
i.e. the compositum M0K lies in Gal (K) and G((M0K)/K) ∼= G(M0/E0) . At the
same time, our argument proves that F is dense in Fπ,ν(p) with respect to the
topology induced by ν(p) , K has a unique valuation κ(π) extending w(π) ,
and (M0K)t(π)′ ∈ Gal(Kκ(π)) with G((M0K)t(π)′/Kκ(π)) ∼= Hπ , for any prolongation
t(π)′ of t(π) on M0K . Let now M(K) = {κ(p): p ∈ P} be a system of valuations
of K complementing κ(π) so that κ(p) extends w(p) or the normalized p -adic
valuation of Q , depending on whether or not p ∈ (P(G) \ {π}) . Applying [Ch3,
Theorem 2.2] to K , P and M(K) , and using [Ch3, Lemmas 3.1 and 3.2], one proves
the existence of an extension E of K in E0,sep , such that P(E) = P , M0E := M lies
in Gal (E) , G(M/E) ∼= G , and E possesses a system {v(p): p ∈ P} of valuations
with the following properties:
(1.2) For each p ∈ P , v(p) extends κ(p) , Ev(p) is a PQL-field, Ev(p)(p) is E -
isomorphic to E(p)⊗E Ev(p) , Br (Ev(p))p 6= {0} and Br (Ev(p′))p = {0} , for every
p′ ∈ (P \ {p}) . Moreover, if p ∈ P(G) and v(p)′ is a valuation of M extending
v(p) , then Mv(p)′ ∈ Gal(Ev(p)) , G(Mv(p)′/Ev(p)) ∼= G(M0,w(p)′/E0,w(p)): p 6= π , and
G(Mv(π)′/Ev(π))
∼= Hπ .
Hence, by [Ch3, Theorem 2.1], E is a nonreal strictly PQL-field. As Ev(π) is PQL,
it follows from [Ch2, Theorem 4.1], statements (1.2) and the Brauer-Hasse-Noether
and Albert theorem (in the form of [Ch3, Proposition 1.2]) that M splits an algebra
∆ ∈ d(E) of π -primary dimension if and only if ind (∆) divides the order of Hπ .
In view of the general theory of simple algebras (see [P, Sects. 13.4 and 14.4]), this
proves Theorem 1.1.
2. Divisible and reduced parts of the multiplicative group of an SQL-field
A field E is said to be strictly quasilocal (SQL), if its finite extensions are strictly
PQL. When this holds and E is almost perfect (in the sense of [Ch2, I, (1.8)]), this
Section gives a Galois-theoretic characterization of the maximal divisible subgroup
D(E) of E∗ .
Theorem 2.1. Let E be an almost perfect SQL-field and N1(E) the intersection of
the groups from NG (E) . Then N1(E) = D(E
To prove Theorem 2.1 we need the following lemma.
Lemma 2.2. Let E and M be fields, M ∈ Gal(E) , and let P(M/E) be the set of
prime divisors of [M:E] . Then N(M/E) ⊆ N(M/F) , for every intermediate field F
of M/E . Moreover, if Ep is the fixed field of some Sylow p -subgroup of G(M/E) ,
then N(M/E) = ∩p∈P(M/E)N(M/Ep) .
Proof. It is easily verified that if [F: E] = m , {τj : j = 1, . . . ,m} is the set of E -
embeddings of F into M , and σu is an automorphism of M extending τu , for each
index j , then G(M/E) = ∪mj=1G(M/F)σ
j . This yields N
E (γ) = N
j=1 σ
j (γ)) ,
for any γ ∈ M∗ , which proves that N(M/F) ⊆ N(M/E) . We show that ∩p∈P(M/E)
N(M/Ep) := N0(M/E) ⊆ N(M/E) . Take an element α ∈ N0(M/E) and put [Ep: E]
= mp , for each p ∈ P(M/E) . It is known that p 6 |mp , for any p ∈ P(M/E) , which
implies consecutively that g.c.d.(mp: p ∈ P(M/E)) = 1 , α ∈ E∗ and αmp ∈ N(M/E) ,
for every p ∈ P(M/E) . Thus it turns out that α ∈ N(M/E) , so Lemma 2.2 is proved.
Proof of Theorem 2.1. The inclusion D(E∗) ⊆ N1(E) is obvious, so our objective is
to prove the converse. We show that N1(E) ⊆ N1(E)p
, for every p ∈ P and n ∈ N .
It is clearly sufficient to consider the special case where p ∈ Π(E) . Then there is a
finite extension Fp of E in Esep , such that Br (Fp)p 6= {0} and p 6 |[Fp: E] . Note
also that each finite extension of Fp in E
∗ is included in a field F̃p ∈ Gal(E) , so it
follows from Lemma 2.2 that N1(E) ⊆ N1(Fp) . At the same time, the norm mapping
E clearly induces homomorphisms D(F
p) → D(E∗) and N1(Fp) → Np(E) (for
the latter, see [L, Ch. VIII, Sect. 5]) as well as an automorphism of the maximal
p -divisible subgroup of E∗ . These observations allow one to assume additionally
that p ∈ P(E) (and Br (E)p 6= {0} ). If p = char(E) , our assertion is implied by
[Ch2, Lemma 8.4], so we turn to the case of p 6= char(E) . Let εp be a primitive
p -th root of unity in Esep . Then [E(εp): E] divides p− 1 (cf. [L, Ch. VIII, Sect.
3]), whence our considerations reduce further to the case in which εp ∈ E . Now
the proof of Theorem 2.1 is completed by applying the following two lemmas. Before
stating them, let us recall that the character group C(Y/E) of G(Y/E) is an abelian
torsion group, for every Galois extension Y/E (see [K, Ch. 7, Sect. 5]). Therefore,
by [F, Theorem 24.5], the divisible part D(Y/E) of C(Y/E) is a direct summand
in C(Y/E) , i.e. C(Y/E) is isomorphic to the direct sum D(Y/E)⊕ R(Y/E) , where
R(Y/E) ∼= C(Y/E)/D(Y/E) is a maximal reduced subgroup of C(Y/E) .
Lemma 2.3. Let E be a p -quasilocal field containing a primitive p -th root of unity
ε , for some prime p ∈ P(E) . Assume also that rp(E) is the group of roots of unity
in E of p -primary degrees. Then:
(i) R(E(p)/E) = {0} if and only if Br (E)p = {0} or rp(E) is infinite;
(ii) If Br (E)p 6= {0} , d is the dimension of pBr(E) as a vector space over the
field Fp with p elements, and Rp(E) is of order p
µ , for some µ ∈ N , then
D(E(p)/E) = pµC(E(p)/E) and R(E(p)/E) is presentable as a direct sum of cyclic
groups of order pµ , indexed by a set of cardinality d .
Proof. Statement (i) is a well-known consequence of Kummer theory, the Merkurjev-
Suslin theorem and Galois cohomology (see [MS, (11.5), (16.1)] and [S, Ch. I, 3.4
and 4.2]), so we assume further that d > 0 and rp(E) is of order p
µ , for some
µ ∈ N . We first show that pµC(E(p)/E) is divisible. Fix a primitive pµ -th root of
unity εµ in E and a subset S = {εn: n ∈ N, n ≥ µ+ 1} of E(p) so that εpn+1 = εn ,
for each index n . Also, let E∞ = E(S) and En = E(εn) , for every integer n > µ .
Suppose first that p > 2 or µ ≥ 2 . Then E∞/E is a Zp -extension and En is the
unique subextension of E in E∞ of degree p
n−µ . Since Zp is a projective profinite
group (see [G, Theorem 1]), this enables one to deduce from Galois theory that E(p)
possesses a subfield E′ such that E′ ∩ E∞ = E and E∞E′ = E(p) . In addition, it
becomes clear that C(E(p)/E) is isomorphic to the direct sum Z(p∞)⊕ C(F/E) ,
where Z(p∞) is the quasicyclic p -group (identified with the character group of Zp )
and F is the maximal abelian extension of E in E′ . Let now Φ be a cyclic extension
of E in F . As E is p -quasilocal, an element βn ∈ E∗n lies in N((ΦEn)/En) , for an
arbitrary integer n ≥ µ , if and only if the norm NEnE (βn) ∈ N(Φ/E) . In particular,
this proves that εµ ∈ N(Φ/E) if and only if εn ∈ N((ΦEn)/En) . In view of the
results of [FSS, Sect. 2], this means that pµC(F/E) = pnC(F/E) , for every index
n ≥ µ . In other words, C(F/E) is divisible, so pµC(E(p)/E) has the same property,
as claimed.
Our objective now is to prove the divisibility of 2C(E(2)/E) , under the hypothesis
that p = 2 and µ = 1 . If E∞ = E(
−1) = E2 , this is contained in [FSS, Proposi-
tion 2], so we assume further that E∞ 6= E2 , i.e. E2 = Eν 6= Eν+1 , for some in-
teger ν ≥ 2 . Let R be a cyclic extension of E in E(2) . By Albert’s theorem
(see [FSS, Sect. 2]), we have C(R/E) ⊂ 2C(E(2)/E) if and only if −1 ∈ N(R/E) .
We show that if C(R/E) ⊂ 2C(E(2)/E) , then εν ∈ N((RE2)/E2) . Since NE2E (εν)
equals 1 or −1 , this follows from [Ch2, Lemma 4.2 (iii)] in case R ∩ E2 = E .
Suppose further that E2 ⊆ R and fix a generator σ of G(F/E) . Applying [Ch2,
Theorem 4.1] one obtains that the cyclic E2 -algebra (R/E2, σ
2, εν) is similar
to (R/E, σ, c)⊗E E2 , for some c ∈ E∗ . In view of [P, Sect. 14.7, Proposition
b], this ensures that (R/E2, σ
2, εν) ∼= (R/E2, σ2, c) as E2 -algebras. Therefore, [P,
Sect. 15.1, Proposition b] yields cε−1ν ∈ N(R/E2) and c2N
E (εν) ∈ N(R/E) . Since
−1 ∈ N(R/E) , this means that c2 ∈ N(R/E) as well. Observing now that the core-
striction homomorphism of Br (E2) into Br (E) maps the similarity class Ac of
(R/E2, σ
2, c) into the similarity class of (R/E, σ, c2) (cf. [T, Theorem 2.5]), one con-
cludes that Ac lies in the kernel Ker E2/E of this homomorphism. At the same
time, by [Ch2, Lemma 4.2 (i)], we have Br (E2)2 ∩KerE2/E = {0} . In particu-
lar, Ac = 0 , so it follows from [P, Sect. 15.1, Proposition b] that εν ∈ N(R/E2) ,
as claimed. The obtained result, combined with [AFSS, Theorem 3], implies
that 2C(E(2)/E) = 2νC(E(2)/E) = 4C(E(2)/E) , which proves the divisibility of
2C(E(2)/E) .
In order to complete the proof of Lemma 2.3, it remains to be seen that the quotient
group C(E(p)/E)/pµC(E(p)/E) is isomorphic to the direct sum of cyclic groups of
order pµ , indexed by a set I of cardinality d (see [F, Theorem 24.5]). Since, by Kum-
mer’s theory, pµ−1C(E(p)/E) includes the group Xp(E) = {χ ∈ C(E(p)/E): pχ = 0} ,
it is sufficient to show that (Xp(E) + p
µC(E(p)/E))/pµC(E(p)/E) := Xp(E) is iso-
morphic to pBr(E) . Applying Albert’s theorem and elementary properties of symbol
E -algebras, and taking into account that Xp(E) ∼= Xp(E)/(Xp(E) ∩ pµC(E(p)/E)) ,
one obtains that Xp(E) ∼= E∗/N(Eµ+1/E) . At the same time, the cyclicity of Eµ+1/E
ensures that E∗/N(Eµ+1/E) ∼= Br(Eµ+1/E) (cf. [P, Sect. 15.1, Proposition b]), and
by the p -quasilocal property of E , we have Br (Eµ+1/E) = pBr(E) , so our proof is
complete.
Lemma 2.4. Let E be a p -quasilocal field containing a primitive p -th root
of unity ε , and such that Br (E)p 6= {0} , and let Ωp(E) be the set of finite
abelian extensions of E in E(p) . Then the intersection Np(E) of the norm groups
N(M/E): M ∈ Ωp(E) , coincides with the maximal p -divisible subgroup of E∗ .
Proof. It is clearly sufficient to show that Np(E) ⊆ N1(E)p
, for each n ∈ N . Denote
by rp(E) the group of roots of unity in E of p -primary degrees and fix an
algebra D ∈ d(E) of index p (the existence of D is guaranteed by the assumption
that Br (E)p 6= {0} and the Merkurjev-Suslin theorem [MS, (16.1)]). As E is p -
quasilocal, one obtains from Kummer theory that, for each c ∈ E∗ \ E∗p , there is
c′ ∈ E∗ \ E∗p , such that D is E -isomorphic to the symbol E -algebra Aε(c, c′; E) .
Hence, c 6∈ N(Ec′/E) , where Ec′ is the extension of E in E(p) obtained by
adjunction of a p -th root of c′ (see [P, Sect. 15.1, Proposition b]). The obtained
result implies Np(E) ⊆ E∗p . Arguing in a similar manner and applying general
properties of cyclic E -algebras (cf. [P, Sect. 15.1, Corollary b]), one deduces that
Np(E) ⊆ E∗p
, if E contains a primitive pm -th root of unity (and proves the
lemma in case rp(E) is infinite). Suppose further that rp(E) is of order p
µ , for
some µ ∈ N , and take C(E(p)/E) , D(E(p)/E) and R(E(p)/E) as in Lemma 2.3.
Also, let c ∈ (E∗pm ∩Np(E)) , for some integer m ≥ µ . We prove Lemma 2.4 by
showing that c ∈ (E∗pm+µ ∩Np(E)p
) . Fix a primitive pµ -th root of unity δµ ∈ E ,
take an element cm ∈ E∗ so that cp
m = c , and for any λ ∈ E∗ , denote by Eλ
the extension of E in E(p) obtained by adjunction of a pµ -th root of λ . Using
again [P, Sect. 15.1, Corollary b], one obtains that cm ∈ N(F/E) whenever F is an
intermediate field of a Zp -extension of E . Hence, by Albert’s theorem (see [FSS,
Sect. 2]), Kummer theory and the basic properties of symbol E -algebras of dimension
p2µ , N(Eδµ)/E) ⊆ N(Ecm)/E) . Since Br (E)p 6= {0} , E admits (one-dimensional)
local p -class field theory [Ch4, Theorem 3.1], which means that Ecm ⊆ Eδµ . In
view of Kummer theory, the obtained result yields cm = δ
m , for some k ∈ N ,
tm ∈ E∗ . Therefore, we have tp
m = (t
pm = c . Let now M be an arbitrary
finite abelian extension of E in E(p) . Then it follows from Galois theory and
Lemma 2.3 that M is a subfield of the compositum of finitely many cyclic extensions
Di, i ∈ I , and Rj , j ∈ J , of E in E(p) , such that [Rj : E] = pµ , for every j ∈ J , and
Di ⊆ ∆i , where ∆i is a Zp -extension of E in E(p) , for each index i . Hence, by
[Ch4, Theorem 3.1], cp
m ∈ N(M̃/E) ⊆ N(M/E) , i.e. cp
m ∈ Np(E) . Replacing cm by
tm and arguing as above, one obtains that t
m ∈ Np(E) and c ∈ Np(E)p
. Thus
the assertion that c ∈ (E∗pm+µ ∩Np(E)p
) is proved. It is now easy to see that
Np(E) = Np(E)
pn , for any n ∈ N , as required by Lemma 2.4.
Remark 2.5. Analyzing the proof of Theorem 2.1, one obtains that its conclu-
sion remains valid, if E is a quasilocal perfect field and cd p(GE) 6= 1 , for all
p ∈ (P \ char(E)) . One also proves that R(E∗) ∼= Φ(E)⊕ R0(E) , where Φ(E) is a
torsion-free group and R0(E) is the direct sum of the groups rp(E) , indexed by
those p ∈ Π(E) , for which rp(E) is nontrivial and finite. Moreover, it turns out that
if Br (E)p 6= {0} , for every p ∈ (Π(E) \ char(E)) , then D(E∗) equals the intersec-
tion of the groups N(M/E) , defined over all M ∈ Gal(E) , such that G(M/E) lies
in any fixed class χ of finite groups, which is closed under the formation of sub-
groups, homomorphic images and finite direct products, and which contains all finite
metabelian groups of orders not divisible by any p ∈ (P \Π(E)) .
3. The reduced part of the Brauer group of an equicharacteristic Henselian
valued absolutely stable field with a totally indivisible value group
In what follows, our notation agrees with that of Lemma 2.3 and its proof. For
each Henselian valued field (K, v) , K̂ and v(K) denote the residue field and
the value group of (K, v) , respectively. In this Section we announce the following
characterization of the reduced components of the Brauer groups of the fields pointed
out in its title:
Theorem 3.1. An abelian torsion group T is isomorphic to the reduced part of
Br (K) , for some absolutely stable field K = K(T) with a Henselian valuation v
such that char (K) = char(K̂) and the value group v(K) is totally indivisible (i.e.
v(K) 6= pv(K) , for every p ∈ P ), if and only if the p -component Tp of T is
presentable as a direct sum of cyclic groups of one and the same order pnp , for
each p ∈ P .
The rest of this Section is devoted to the proof of the necessity in Theorem 3.1
(the sufficiency will be proved elsewhere in a more general situation covering the
case where char (K) 6= char(K̂) , v is Henselian and discrete, and K̂ is perfect). We
begin with a description of some known basic relations between Br (K)p , Br (K̂)p
and C(K̂(p)/K̂) (proved for convenience of the reader).
Lemma 3.2. Let (K, v) be a Henselian valued field with v(K) 6= pv(K) , for
some p ∈ P , and let π be an element of K∗ of value v(π) 6∈ pv(K) . For each
χ ∈ C(K̂(p)/K̂) , denote by L̃χ the extension of K̂ in K̂sep corresponding by Ga-
lois theory to the kernel Ker (χ) , and by σ̃χ the generator of G(L̃χ/K̂) satis-
fying the equality χ(σ̃χ) = (1/[Lχ: K̂]) +Q/Z . Assume also that Lχ is the iner-
tial lift of L̃χ (over K̂ ) in Ksep , ηχ is the canonical isomorphism of G(Lχ/K)
on G(L̃χ/K̂) , and σχ is the preimage of σ̃χ in G(Lχ/K) with respect to
ηχ . Then the mapping Wπ of C(K̂(p)/K̂) into Br (K)p defined by the rule
Wπ(χ) = [(Lχ/K, σχ, π)]: χ ∈ C(K̂(p)/K̂) , is an injective group homomorphism.
Proof. It is clearly sufficient to show that [(Lχ1+χ2/K, σχ1+χ2 , π)] = [(Lχ1/L, σχ1 , π)]
+[(Lχ2/K, σχ2 , π)] , where χ1, χ2 ∈ C(K̂(p)/K̂) , in each of the following special
cases:
(3.1) (i) χ2 ∈ 〈χ1〉 , i.e. Lχ2 ⊆ Lχ1 ;
(ii) 〈χ1〉 ∩ 〈χ2〉 = {0} , i.e. Lχ1 ∩ Lχ2 = K .
In case (3.1) (i), this follows from the general properties of cyclic algebras (cf. [P,
Sect. 15.1, Corollary b and Proposition b]). Assuming that Lχ1 ∩ Lχ2 = K and
[Lχ1 : K] ≥ [Lχ2 : K] , one obtains that the K -algebras (Lχ1/K, σχ2 , π)⊗K (Lχ2/K, σχ2 ,
π) and (Lχ1+χ2/K, σχ1+χ2 , π)⊗K (Lχ2/K, σχ2 , 1) are isomorphic, which completes
our proof.
With assumptions being as in the lemma, let v(K) 6= pv(K) , for some prime
p 6= char(K̂) , and let Bp be a basis and n(p) the dimension of v(K)/pv(K)
as a vector space over Fp . Denote by C(K̂(p)/K̂)
n(p) the direct sum of some
isomorphic copies of C(K̂(p)/K̂) , indexed by Bp . Fix a linear ordering ≤ on Bp
and put Jp = {(cp, dp) ∈ (Bp × Bp): cp < dp} , and in case n(p) ≥ 2 , take a direct
sum Rp(K̂)
Jp of isomorphic copies of Rp(K̂) , indexed by Jp . Then the Ostrowski-
Draxl theorem, the Jacob-Wadsworth decomposition lemmas (see [JW]) and Lemma
3.2 imply the following variant of the Scharlau-Witt theorem [Sch]:
(3.2) (i) Br (K)p ∼= Br(K̂)p ⊕ C(K̂(p)/K̂)n(p) unless n(p) ≥ 2 and Rp(K̂) 6= {1} ;
(ii) Br (K)p ∼= Br(K̂)p ⊕ C(K̂(p)/K̂)n(p) ⊕ Rp(K̂)Jp , if n(p) ≥ 2 and Rp(K̂) 6= {1} .
Suppose now that K is a Henselian valued absolutely stable field such that v(K)
is totally indivisible. By [Ch2, Proposition 2.3], K̂ is quasilocal, so the necessity in
Theorem 3.1 can be deduced from (3.2), the divisibility of Br (F)q , for any field F of
characteristic q > 0 (Witt’s theorem, see [Dr, Sect. 15]), and the following lemma.
Before stating it, note that the extension of E in Esep obtained by adjunction of a
primitive p -th root of unity, for some p ∈ P , is cyclic of degree dividing p− 1 (cf.
[L, Ch. VIII, Sect. 3]).
Lemma 3.3. Let E be a quasilocal field not containing a primitive p -th root of unity,
for a given prime p 6= char(E) , and let ε be such a root in Esep . Fix a generator
ϕ of G(E(ε)/E) , take an integer s so as to satisfy the equality ϕ(ε) = εs , and put
Bs = {b ∈ pBr(E(ε)): ϕ(b) = sb} . Then:
(i) R(E(p)/E) = {0} if and only if rp(E(ε)) is infinite or the group Bs is trivial;
(ii) If Bs is nontrivial and of dimension d as an Fp -vector space, and if rp(E(ε))
is of order pµ , for some µ ∈ N , then D(E(p)/E) = pµC(E(p)/E) and R(E(p)/E)
is a direct sum of cyclic groups of order pµ , indexed by a set of cardinality d .
Proof. Let Λ = {λ ∈ E(ε)∗: ϕ(λ)λ−s ∈ E(ε)∗p} . It is known (Albert, see [A, Ch. IX,
Theorem 15]) that an extension L of E in Esep is cyclic of degree p if and only if
L(ε) is generated over E(ε) by a p -th root of an element λ0 ∈ (Λ \ E(ε)∗p) . Hence,
the Kummer isomorphism of Xp(E(ε)) on E(ε)
∗/E(ε)∗p induces an isomorphism
of Xp(E) on Λ/E(ε)
∗p . Observe also that the similarity class of the symbol E(ε) -
algebra Aε(λ1, λ2; E(ε)) lies in Bi whenever λj ∈ Λ , j = 1, 2 . Conversely, by the
p -quasilocal property of E(ε) , each element of Bs is presented by a symbol E(ε) -
algebra determined by a pair of elements of Λ . The noted results enable one to prove
the lemma arguing as in the proof of Lemma 2.3.
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Ivan CHIPCHAKOV
Institute of Mathematics and Informatics
Bulgarian Academy of Sciences
Acad. G. Bonchev Str., bl. 8
1113 SOFIA, Bulgaria

  Let $E$ be a primarily quasilocal field, $M/E$ a finite Galois extension and
$D$ a central division $E$-algebra of index divisible by $[M\colon E]$. In
addition to the main result of Part I, this part of the paper shows that if the
Galois group $G(M/E)$ is not nilpotent, then $M$ does not necessarily embed in
$D$ as an $E$-subalgebra. When $E$ is quasilocal, we find the structure of the
character group of its absolute Galois group; this enables us to prove that if
$E$ is strictly quasilocal and almost perfect, then the divisible part of the
multiplicative group $E ^{\ast}$ equals the intersection of the norm groups of
finite Galois extensions of $E$.

Introduction
This paper is a continuation of [Ch2]. Let E be a field, Esep a separable closure of
E , E∗ the multiplicative group of E , d(E) the class of finite-dimensional central
division E -algebras, Br (E) the Brauer group of E , Gal (E) the set of finite Galois
extensions of E in Esep , and NG (E) the set of norm groups N(M/E): M ∈ Gal(E) .
Let also P be the set of prime numbers, Br (E)p the p -component of Br (E) , for each
p ∈ P , P(E) the set of those p ∈ P , for which E is properly included in its maximal
p -extension E(p) (in Esep ), and Π(E) the set of all p
′ ∈ P , for which the absolute
Galois group GE = G(Esep/E) is of nonzero cohomological p
′ -dimension cd p′(GE) .
Recall that E is p -quasilocal, for a given p ∈ P , if p 6∈ P(E) or the relative
Brauer group Br (F/E) equals the subgroup pBr(E) = {bp ∈ Br(E): pbp = 0} , for
every cyclic extension F of E of degree p . The field E is said to be primarily
quasilocal (PQL), if it is p -quasilocal, for all p ∈ P ; E is called quasilocal, if its
finite extensions are PQL. We say that E is a strictly PQL-field, if it is PQL and
Br (E)p 6= {0} , for every p ∈ P(E) (see [Ch4, Corollary 3.7], for a characterization
of the SQL-property). The purpose of this note is to complement the main result of
[Ch2] and to shed light on the structure of Br (K) , where K is an absolutely stable
field (in the sense of E. Brussel) with a Henselian valuation whose value group is
totally indivisible.
1. Subfields of central division algebras over PQL-fields
Assume that E is a PQL-field and M ∈ Gal(E) , such that G(M/E) is nilpotent, and
let R be an intermediate field of M/E . It follows from Galois theory, [Ch2, Theorem
4.1 (iii)], the Burnside-Wielandt characterization of nilpotent finite groups, and the
general properties of central division algebras and their Schur indices (see [KM, Ch.
6, Sect. 2] and [P, Sects. 13.4 and 14.4]) that R embeds as an E -subalgebra in each
∗ Partially supported by Grant MI-1503/2005 of the Bulgarian Foundation for Scien-
tific Research.
http://arxiv.org/abs/0704.1557v1
algebra D ∈ d(E) of index divisible by [R: E] . In this Section, we demonstrate the
optimality of this result in the class of strictly PQL-fields.
Theorem 1.1. For each nonnilpotent finite group G , there exists a strictly PQL-
field E = E(G) with Br (E) ∼= Q/Z and Gal (E) containing an element M such that
G(M/E) ∼= G and M does not embed as an E -subalgebra in any ∆ ∈ d(E) of index
equal to [M:E] .
Proof. Let P(G) be the set of prime divisors of the order of G . Our argument relies
on the existence (cf. [Ch1, Sect. 4]) of an algebraic number field E0 , such that there
is M0 ∈ Gal(E0) with G(M0/E0) ∼= G , and E0 possesses a system of (real-valued)
valuations {w(p): p ∈ P(G)} satisfying the following conditions:
(1.1) (i) w(p) extends the normalized p -adic valuation (in the sense of [CF, Ch.
VII]) of the field Q of rational numbers, for each p ∈ P(G) ;
(ii) The completion M0,w(p)′ lies in Gal (E0,w(p)) and G(M0,w(p)′/E0,w(p)) is isomor-
phic to a Sylow p -subgroup of G(M0/E0) , whenever p ∈ P(G) and w(p)′ extends
w(p) on M0 .
Let Ap be the maximal abelian p -extension of E0 in M0 , for each p ∈ P(G) . It
follows from the nonnilpotency of G , the Burnside-Wielandt theorem and Galois
theory that M0/E0 has an intermediate field F 6= E0 such that F ∩ Ap = E , for
all p ∈ P(G) (see [Ch1, Sect. 4]). Fix a prime divisor π of [F: E0] as well as Sylow
π -subgroups Hπ ∈ SylπG(M0/F) and Gπ ∈ SylπG(M0/E0) with Hπ ⊂ Gπ , and
denote by Fπ and Eπ the maximal extensions of E0 in M0 fixed by Hπ and
Gπ , respectively. It is clear from (1.1) (ii) and [CF, Ch. II, Theorem 10.2] that Eπ
has a valuation t(π) extending w(π) so that M0 ⊗Eπ Eπ,t(π) is a field. This means
that t(π) is uniquely (up-to an equivalence) extendable to a valuation µ(π) of M0 ,
and allows us to identify E0,w(π) with Eπ,t(π) . Using repeatedly the Grunwald-Wang
theorem [W] and the normality of maximal subgroups of finite π -groups (cf. [L, Ch. I,
Sect. 6]), one also obtains that there is a finite extension K of E0 in E0(π) , such that
K⊗E0 E0,w(π) is a field isomorphic to Fπ,ν(π) as an E0 -algebra, where ν(π) is the
valuation of Fπ induced by µ(π) . These observations indicate that K ∩M0 = E0 ,
i.e. the compositum M0K lies in Gal (K) and G((M0K)/K) ∼= G(M0/E0) . At the
same time, our argument proves that F is dense in Fπ,ν(p) with respect to the
topology induced by ν(p) , K has a unique valuation κ(π) extending w(π) ,
and (M0K)t(π)′ ∈ Gal(Kκ(π)) with G((M0K)t(π)′/Kκ(π)) ∼= Hπ , for any prolongation
t(π)′ of t(π) on M0K . Let now M(K) = {κ(p): p ∈ P} be a system of valuations
of K complementing κ(π) so that κ(p) extends w(p) or the normalized p -adic
valuation of Q , depending on whether or not p ∈ (P(G) \ {π}) . Applying [Ch3,
Theorem 2.2] to K , P and M(K) , and using [Ch3, Lemmas 3.1 and 3.2], one proves
the existence of an extension E of K in E0,sep , such that P(E) = P , M0E := M lies
in Gal (E) , G(M/E) ∼= G , and E possesses a system {v(p): p ∈ P} of valuations
with the following properties:
(1.2) For each p ∈ P , v(p) extends κ(p) , Ev(p) is a PQL-field, Ev(p)(p) is E -
isomorphic to E(p)⊗E Ev(p) , Br (Ev(p))p 6= {0} and Br (Ev(p′))p = {0} , for every
p′ ∈ (P \ {p}) . Moreover, if p ∈ P(G) and v(p)′ is a valuation of M extending
v(p) , then Mv(p)′ ∈ Gal(Ev(p)) , G(Mv(p)′/Ev(p)) ∼= G(M0,w(p)′/E0,w(p)): p 6= π , and
G(Mv(π)′/Ev(π))
∼= Hπ .
Hence, by [Ch3, Theorem 2.1], E is a nonreal strictly PQL-field. As Ev(π) is PQL,
it follows from [Ch2, Theorem 4.1], statements (1.2) and the Brauer-Hasse-Noether
and Albert theorem (in the form of [Ch3, Proposition 1.2]) that M splits an algebra
∆ ∈ d(E) of π -primary dimension if and only if ind (∆) divides the order of Hπ .
In view of the general theory of simple algebras (see [P, Sects. 13.4 and 14.4]), this
proves Theorem 1.1.
2. Divisible and reduced parts of the multiplicative group of an SQL-field
A field E is said to be strictly quasilocal (SQL), if its finite extensions are strictly
PQL. When this holds and E is almost perfect (in the sense of [Ch2, I, (1.8)]), this
Section gives a Galois-theoretic characterization of the maximal divisible subgroup
D(E) of E∗ .
Theorem 2.1. Let E be an almost perfect SQL-field and N1(E) the intersection of
the groups from NG (E) . Then N1(E) = D(E
To prove Theorem 2.1 we need the following lemma.
Lemma 2.2. Let E and M be fields, M ∈ Gal(E) , and let P(M/E) be the set of
prime divisors of [M:E] . Then N(M/E) ⊆ N(M/F) , for every intermediate field F
of M/E . Moreover, if Ep is the fixed field of some Sylow p -subgroup of G(M/E) ,
then N(M/E) = ∩p∈P(M/E)N(M/Ep) .
Proof. It is easily verified that if [F: E] = m , {τj : j = 1, . . . ,m} is the set of E -
embeddings of F into M , and σu is an automorphism of M extending τu , for each
index j , then G(M/E) = ∪mj=1G(M/F)σ
j . This yields N
E (γ) = N
j=1 σ
j (γ)) ,
for any γ ∈ M∗ , which proves that N(M/F) ⊆ N(M/E) . We show that ∩p∈P(M/E)
N(M/Ep) := N0(M/E) ⊆ N(M/E) . Take an element α ∈ N0(M/E) and put [Ep: E]
= mp , for each p ∈ P(M/E) . It is known that p 6 |mp , for any p ∈ P(M/E) , which
implies consecutively that g.c.d.(mp: p ∈ P(M/E)) = 1 , α ∈ E∗ and αmp ∈ N(M/E) ,
for every p ∈ P(M/E) . Thus it turns out that α ∈ N(M/E) , so Lemma 2.2 is proved.
Proof of Theorem 2.1. The inclusion D(E∗) ⊆ N1(E) is obvious, so our objective is
to prove the converse. We show that N1(E) ⊆ N1(E)p
, for every p ∈ P and n ∈ N .
It is clearly sufficient to consider the special case where p ∈ Π(E) . Then there is a
finite extension Fp of E in Esep , such that Br (Fp)p 6= {0} and p 6 |[Fp: E] . Note
also that each finite extension of Fp in E
∗ is included in a field F̃p ∈ Gal(E) , so it
follows from Lemma 2.2 that N1(E) ⊆ N1(Fp) . At the same time, the norm mapping
E clearly induces homomorphisms D(F
p) → D(E∗) and N1(Fp) → Np(E) (for
the latter, see [L, Ch. VIII, Sect. 5]) as well as an automorphism of the maximal
p -divisible subgroup of E∗ . These observations allow one to assume additionally
that p ∈ P(E) (and Br (E)p 6= {0} ). If p = char(E) , our assertion is implied by
[Ch2, Lemma 8.4], so we turn to the case of p 6= char(E) . Let εp be a primitive
p -th root of unity in Esep . Then [E(εp): E] divides p− 1 (cf. [L, Ch. VIII, Sect.
3]), whence our considerations reduce further to the case in which εp ∈ E . Now
the proof of Theorem 2.1 is completed by applying the following two lemmas. Before
stating them, let us recall that the character group C(Y/E) of G(Y/E) is an abelian
torsion group, for every Galois extension Y/E (see [K, Ch. 7, Sect. 5]). Therefore,
by [F, Theorem 24.5], the divisible part D(Y/E) of C(Y/E) is a direct summand
in C(Y/E) , i.e. C(Y/E) is isomorphic to the direct sum D(Y/E)⊕ R(Y/E) , where
R(Y/E) ∼= C(Y/E)/D(Y/E) is a maximal reduced subgroup of C(Y/E) .
Lemma 2.3. Let E be a p -quasilocal field containing a primitive p -th root of unity
ε , for some prime p ∈ P(E) . Assume also that rp(E) is the group of roots of unity
in E of p -primary degrees. Then:
(i) R(E(p)/E) = {0} if and only if Br (E)p = {0} or rp(E) is infinite;
(ii) If Br (E)p 6= {0} , d is the dimension of pBr(E) as a vector space over the
field Fp with p elements, and Rp(E) is of order p
µ , for some µ ∈ N , then
D(E(p)/E) = pµC(E(p)/E) and R(E(p)/E) is presentable as a direct sum of cyclic
groups of order pµ , indexed by a set of cardinality d .
Proof. Statement (i) is a well-known consequence of Kummer theory, the Merkurjev-
Suslin theorem and Galois cohomology (see [MS, (11.5), (16.1)] and [S, Ch. I, 3.4
and 4.2]), so we assume further that d > 0 and rp(E) is of order p
µ , for some
µ ∈ N . We first show that pµC(E(p)/E) is divisible. Fix a primitive pµ -th root of
unity εµ in E and a subset S = {εn: n ∈ N, n ≥ µ+ 1} of E(p) so that εpn+1 = εn ,
for each index n . Also, let E∞ = E(S) and En = E(εn) , for every integer n > µ .
Suppose first that p > 2 or µ ≥ 2 . Then E∞/E is a Zp -extension and En is the
unique subextension of E in E∞ of degree p
n−µ . Since Zp is a projective profinite
group (see [G, Theorem 1]), this enables one to deduce from Galois theory that E(p)
possesses a subfield E′ such that E′ ∩ E∞ = E and E∞E′ = E(p) . In addition, it
becomes clear that C(E(p)/E) is isomorphic to the direct sum Z(p∞)⊕ C(F/E) ,
where Z(p∞) is the quasicyclic p -group (identified with the character group of Zp )
and F is the maximal abelian extension of E in E′ . Let now Φ be a cyclic extension
of E in F . As E is p -quasilocal, an element βn ∈ E∗n lies in N((ΦEn)/En) , for an
arbitrary integer n ≥ µ , if and only if the norm NEnE (βn) ∈ N(Φ/E) . In particular,
this proves that εµ ∈ N(Φ/E) if and only if εn ∈ N((ΦEn)/En) . In view of the
results of [FSS, Sect. 2], this means that pµC(F/E) = pnC(F/E) , for every index
n ≥ µ . In other words, C(F/E) is divisible, so pµC(E(p)/E) has the same property,
as claimed.
Our objective now is to prove the divisibility of 2C(E(2)/E) , under the hypothesis
that p = 2 and µ = 1 . If E∞ = E(
−1) = E2 , this is contained in [FSS, Proposi-
tion 2], so we assume further that E∞ 6= E2 , i.e. E2 = Eν 6= Eν+1 , for some in-
teger ν ≥ 2 . Let R be a cyclic extension of E in E(2) . By Albert’s theorem
(see [FSS, Sect. 2]), we have C(R/E) ⊂ 2C(E(2)/E) if and only if −1 ∈ N(R/E) .
We show that if C(R/E) ⊂ 2C(E(2)/E) , then εν ∈ N((RE2)/E2) . Since NE2E (εν)
equals 1 or −1 , this follows from [Ch2, Lemma 4.2 (iii)] in case R ∩ E2 = E .
Suppose further that E2 ⊆ R and fix a generator σ of G(F/E) . Applying [Ch2,
Theorem 4.1] one obtains that the cyclic E2 -algebra (R/E2, σ
2, εν) is similar
to (R/E, σ, c)⊗E E2 , for some c ∈ E∗ . In view of [P, Sect. 14.7, Proposition
b], this ensures that (R/E2, σ
2, εν) ∼= (R/E2, σ2, c) as E2 -algebras. Therefore, [P,
Sect. 15.1, Proposition b] yields cε−1ν ∈ N(R/E2) and c2N
E (εν) ∈ N(R/E) . Since
−1 ∈ N(R/E) , this means that c2 ∈ N(R/E) as well. Observing now that the core-
striction homomorphism of Br (E2) into Br (E) maps the similarity class Ac of
(R/E2, σ
2, c) into the similarity class of (R/E, σ, c2) (cf. [T, Theorem 2.5]), one con-
cludes that Ac lies in the kernel Ker E2/E of this homomorphism. At the same
time, by [Ch2, Lemma 4.2 (i)], we have Br (E2)2 ∩KerE2/E = {0} . In particu-
lar, Ac = 0 , so it follows from [P, Sect. 15.1, Proposition b] that εν ∈ N(R/E2) ,
as claimed. The obtained result, combined with [AFSS, Theorem 3], implies
that 2C(E(2)/E) = 2νC(E(2)/E) = 4C(E(2)/E) , which proves the divisibility of
2C(E(2)/E) .
In order to complete the proof of Lemma 2.3, it remains to be seen that the quotient
group C(E(p)/E)/pµC(E(p)/E) is isomorphic to the direct sum of cyclic groups of
order pµ , indexed by a set I of cardinality d (see [F, Theorem 24.5]). Since, by Kum-
mer’s theory, pµ−1C(E(p)/E) includes the group Xp(E) = {χ ∈ C(E(p)/E): pχ = 0} ,
it is sufficient to show that (Xp(E) + p
µC(E(p)/E))/pµC(E(p)/E) := Xp(E) is iso-
morphic to pBr(E) . Applying Albert’s theorem and elementary properties of symbol
E -algebras, and taking into account that Xp(E) ∼= Xp(E)/(Xp(E) ∩ pµC(E(p)/E)) ,
one obtains that Xp(E) ∼= E∗/N(Eµ+1/E) . At the same time, the cyclicity of Eµ+1/E
ensures that E∗/N(Eµ+1/E) ∼= Br(Eµ+1/E) (cf. [P, Sect. 15.1, Proposition b]), and
by the p -quasilocal property of E , we have Br (Eµ+1/E) = pBr(E) , so our proof is
complete.
Lemma 2.4. Let E be a p -quasilocal field containing a primitive p -th root
of unity ε , and such that Br (E)p 6= {0} , and let Ωp(E) be the set of finite
abelian extensions of E in E(p) . Then the intersection Np(E) of the norm groups
N(M/E): M ∈ Ωp(E) , coincides with the maximal p -divisible subgroup of E∗ .
Proof. It is clearly sufficient to show that Np(E) ⊆ N1(E)p
, for each n ∈ N . Denote
by rp(E) the group of roots of unity in E of p -primary degrees and fix an
algebra D ∈ d(E) of index p (the existence of D is guaranteed by the assumption
that Br (E)p 6= {0} and the Merkurjev-Suslin theorem [MS, (16.1)]). As E is p -
quasilocal, one obtains from Kummer theory that, for each c ∈ E∗ \ E∗p , there is
c′ ∈ E∗ \ E∗p , such that D is E -isomorphic to the symbol E -algebra Aε(c, c′; E) .
Hence, c 6∈ N(Ec′/E) , where Ec′ is the extension of E in E(p) obtained by
adjunction of a p -th root of c′ (see [P, Sect. 15.1, Proposition b]). The obtained
result implies Np(E) ⊆ E∗p . Arguing in a similar manner and applying general
properties of cyclic E -algebras (cf. [P, Sect. 15.1, Corollary b]), one deduces that
Np(E) ⊆ E∗p
, if E contains a primitive pm -th root of unity (and proves the
lemma in case rp(E) is infinite). Suppose further that rp(E) is of order p
µ , for
some µ ∈ N , and take C(E(p)/E) , D(E(p)/E) and R(E(p)/E) as in Lemma 2.3.
Also, let c ∈ (E∗pm ∩Np(E)) , for some integer m ≥ µ . We prove Lemma 2.4 by
showing that c ∈ (E∗pm+µ ∩Np(E)p
) . Fix a primitive pµ -th root of unity δµ ∈ E ,
take an element cm ∈ E∗ so that cp
m = c , and for any λ ∈ E∗ , denote by Eλ
the extension of E in E(p) obtained by adjunction of a pµ -th root of λ . Using
again [P, Sect. 15.1, Corollary b], one obtains that cm ∈ N(F/E) whenever F is an
intermediate field of a Zp -extension of E . Hence, by Albert’s theorem (see [FSS,
Sect. 2]), Kummer theory and the basic properties of symbol E -algebras of dimension
p2µ , N(Eδµ)/E) ⊆ N(Ecm)/E) . Since Br (E)p 6= {0} , E admits (one-dimensional)
local p -class field theory [Ch4, Theorem 3.1], which means that Ecm ⊆ Eδµ . In
view of Kummer theory, the obtained result yields cm = δ
m , for some k ∈ N ,
tm ∈ E∗ . Therefore, we have tp
m = (t
pm = c . Let now M be an arbitrary
finite abelian extension of E in E(p) . Then it follows from Galois theory and
Lemma 2.3 that M is a subfield of the compositum of finitely many cyclic extensions
Di, i ∈ I , and Rj , j ∈ J , of E in E(p) , such that [Rj : E] = pµ , for every j ∈ J , and
Di ⊆ ∆i , where ∆i is a Zp -extension of E in E(p) , for each index i . Hence, by
[Ch4, Theorem 3.1], cp
m ∈ N(M̃/E) ⊆ N(M/E) , i.e. cp
m ∈ Np(E) . Replacing cm by
tm and arguing as above, one obtains that t
m ∈ Np(E) and c ∈ Np(E)p
. Thus
the assertion that c ∈ (E∗pm+µ ∩Np(E)p
) is proved. It is now easy to see that
Np(E) = Np(E)
pn , for any n ∈ N , as required by Lemma 2.4.
Remark 2.5. Analyzing the proof of Theorem 2.1, one obtains that its conclu-
sion remains valid, if E is a quasilocal perfect field and cd p(GE) 6= 1 , for all
p ∈ (P \ char(E)) . One also proves that R(E∗) ∼= Φ(E)⊕ R0(E) , where Φ(E) is a
torsion-free group and R0(E) is the direct sum of the groups rp(E) , indexed by
those p ∈ Π(E) , for which rp(E) is nontrivial and finite. Moreover, it turns out that
if Br (E)p 6= {0} , for every p ∈ (Π(E) \ char(E)) , then D(E∗) equals the intersec-
tion of the groups N(M/E) , defined over all M ∈ Gal(E) , such that G(M/E) lies
in any fixed class χ of finite groups, which is closed under the formation of sub-
groups, homomorphic images and finite direct products, and which contains all finite
metabelian groups of orders not divisible by any p ∈ (P \Π(E)) .
3. The reduced part of the Brauer group of an equicharacteristic Henselian
valued absolutely stable field with a totally indivisible value group
In what follows, our notation agrees with that of Lemma 2.3 and its proof. For
each Henselian valued field (K, v) , K̂ and v(K) denote the residue field and
the value group of (K, v) , respectively. In this Section we announce the following
characterization of the reduced components of the Brauer groups of the fields pointed
out in its title:
Theorem 3.1. An abelian torsion group T is isomorphic to the reduced part of
Br (K) , for some absolutely stable field K = K(T) with a Henselian valuation v
such that char (K) = char(K̂) and the value group v(K) is totally indivisible (i.e.
v(K) 6= pv(K) , for every p ∈ P ), if and only if the p -component Tp of T is
presentable as a direct sum of cyclic groups of one and the same order pnp , for
each p ∈ P .
The rest of this Section is devoted to the proof of the necessity in Theorem 3.1
(the sufficiency will be proved elsewhere in a more general situation covering the
case where char (K) 6= char(K̂) , v is Henselian and discrete, and K̂ is perfect). We
begin with a description of some known basic relations between Br (K)p , Br (K̂)p
and C(K̂(p)/K̂) (proved for convenience of the reader).
Lemma 3.2. Let (K, v) be a Henselian valued field with v(K) 6= pv(K) , for
some p ∈ P , and let π be an element of K∗ of value v(π) 6∈ pv(K) . For each
χ ∈ C(K̂(p)/K̂) , denote by L̃χ the extension of K̂ in K̂sep corresponding by Ga-
lois theory to the kernel Ker (χ) , and by σ̃χ the generator of G(L̃χ/K̂) satis-
fying the equality χ(σ̃χ) = (1/[Lχ: K̂]) +Q/Z . Assume also that Lχ is the iner-
tial lift of L̃χ (over K̂ ) in Ksep , ηχ is the canonical isomorphism of G(Lχ/K)
on G(L̃χ/K̂) , and σχ is the preimage of σ̃χ in G(Lχ/K) with respect to
ηχ . Then the mapping Wπ of C(K̂(p)/K̂) into Br (K)p defined by the rule
Wπ(χ) = [(Lχ/K, σχ, π)]: χ ∈ C(K̂(p)/K̂) , is an injective group homomorphism.
Proof. It is clearly sufficient to show that [(Lχ1+χ2/K, σχ1+χ2 , π)] = [(Lχ1/L, σχ1 , π)]
+[(Lχ2/K, σχ2 , π)] , where χ1, χ2 ∈ C(K̂(p)/K̂) , in each of the following special
cases:
(3.1) (i) χ2 ∈ 〈χ1〉 , i.e. Lχ2 ⊆ Lχ1 ;
(ii) 〈χ1〉 ∩ 〈χ2〉 = {0} , i.e. Lχ1 ∩ Lχ2 = K .
In case (3.1) (i), this follows from the general properties of cyclic algebras (cf. [P,
Sect. 15.1, Corollary b and Proposition b]). Assuming that Lχ1 ∩ Lχ2 = K and
[Lχ1 : K] ≥ [Lχ2 : K] , one obtains that the K -algebras (Lχ1/K, σχ2 , π)⊗K (Lχ2/K, σχ2 ,
π) and (Lχ1+χ2/K, σχ1+χ2 , π)⊗K (Lχ2/K, σχ2 , 1) are isomorphic, which completes
our proof.
With assumptions being as in the lemma, let v(K) 6= pv(K) , for some prime
p 6= char(K̂) , and let Bp be a basis and n(p) the dimension of v(K)/pv(K)
as a vector space over Fp . Denote by C(K̂(p)/K̂)
n(p) the direct sum of some
isomorphic copies of C(K̂(p)/K̂) , indexed by Bp . Fix a linear ordering ≤ on Bp
and put Jp = {(cp, dp) ∈ (Bp × Bp): cp < dp} , and in case n(p) ≥ 2 , take a direct
sum Rp(K̂)
Jp of isomorphic copies of Rp(K̂) , indexed by Jp . Then the Ostrowski-
Draxl theorem, the Jacob-Wadsworth decomposition lemmas (see [JW]) and Lemma
3.2 imply the following variant of the Scharlau-Witt theorem [Sch]:
(3.2) (i) Br (K)p ∼= Br(K̂)p ⊕ C(K̂(p)/K̂)n(p) unless n(p) ≥ 2 and Rp(K̂) 6= {1} ;
(ii) Br (K)p ∼= Br(K̂)p ⊕ C(K̂(p)/K̂)n(p) ⊕ Rp(K̂)Jp , if n(p) ≥ 2 and Rp(K̂) 6= {1} .
Suppose now that K is a Henselian valued absolutely stable field such that v(K)
is totally indivisible. By [Ch2, Proposition 2.3], K̂ is quasilocal, so the necessity in
Theorem 3.1 can be deduced from (3.2), the divisibility of Br (F)q , for any field F of
characteristic q > 0 (Witt’s theorem, see [Dr, Sect. 15]), and the following lemma.
Before stating it, note that the extension of E in Esep obtained by adjunction of a
primitive p -th root of unity, for some p ∈ P , is cyclic of degree dividing p− 1 (cf.
[L, Ch. VIII, Sect. 3]).
Lemma 3.3. Let E be a quasilocal field not containing a primitive p -th root of unity,
for a given prime p 6= char(E) , and let ε be such a root in Esep . Fix a generator
ϕ of G(E(ε)/E) , take an integer s so as to satisfy the equality ϕ(ε) = εs , and put
Bs = {b ∈ pBr(E(ε)): ϕ(b) = sb} . Then:
(i) R(E(p)/E) = {0} if and only if rp(E(ε)) is infinite or the group Bs is trivial;
(ii) If Bs is nontrivial and of dimension d as an Fp -vector space, and if rp(E(ε))
is of order pµ , for some µ ∈ N , then D(E(p)/E) = pµC(E(p)/E) and R(E(p)/E)
is a direct sum of cyclic groups of order pµ , indexed by a set of cardinality d .
Proof. Let Λ = {λ ∈ E(ε)∗: ϕ(λ)λ−s ∈ E(ε)∗p} . It is known (Albert, see [A, Ch. IX,
Theorem 15]) that an extension L of E in Esep is cyclic of degree p if and only if
L(ε) is generated over E(ε) by a p -th root of an element λ0 ∈ (Λ \ E(ε)∗p) . Hence,
the Kummer isomorphism of Xp(E(ε)) on E(ε)
∗/E(ε)∗p induces an isomorphism
of Xp(E) on Λ/E(ε)
∗p . Observe also that the similarity class of the symbol E(ε) -
algebra Aε(λ1, λ2; E(ε)) lies in Bi whenever λj ∈ Λ , j = 1, 2 . Conversely, by the
p -quasilocal property of E(ε) , each element of Bs is presented by a symbol E(ε) -
algebra determined by a pair of elements of Λ . The noted results enable one to prove
the lemma arguing as in the proof of Lemma 2.3.
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Ivan CHIPCHAKOV
Institute of Mathematics and Informatics
Bulgarian Academy of Sciences
Acad. G. Bonchev Str., bl. 8
1113 SOFIA, Bulgaria
