1.2. SIMPLICITY, ISOTYPICITY, AND ENDOMORPHISM ALGEBRAS 19
by a unique central division algebra over K. In this sense, Br(K) is an abelian
group structure on the set of isomorphism classes of such division algebras.
1.2.4.2. Example. The computation of the Brauer group of a number field in-
volves computing the Brauer groups of local fields, so we now clear up any possible
confusion concerning sign conventions in the description of Brauer groups for non-
archimedean local fields. Upon choosing a separable closure Ks of an arbitrary
field K, there are two natural procedures to define a functorial group isomorphism
Br(K) H2(Ks/K, Ks ×): a conceptual method via non-abelian cohomology as in
[107, Ch. X, §5] and an explicit method via crossed-product algebras. By [107,
Ch. X, §5, Exer. 2], these procedures are negatives of each other. We use the
conceptual method of non-abelian cohomology, but we do not need to make that
method explicit here and so we refer the interested reader to [107] for the details.
Let K be a non-archimedean local field with residue field κ and let Kun denote
its maximal unramified subextension in Ks (with κ the residue field of
Kun).
It
is known from local class field theory that the natural map H2(Kun/K,
Kun×)

H2(Ks/K, Ks ×) is an isomorphism, and the normalized valuation mapping
Kun×

Z induces an isomorphism
H2(Kun/K, Kun×) H2(Kun/K,
Z)
δ
H1(Gal(Kun/K),
Q/Z)
=
H1(Gal(κ/κ),
Q/Z).
There now arises the question of choice of topological generator for Gal(κ/κ): arith-
metic or geometric Frobenius? We choose to work with arithmetic Frobenius. (In
[103, §1.1] and [107, Ch. XIII, §3] the arithmetic Frobenius generator is also used.)
Via evaluation on the chosen topological generator, our conventions lead to a
composite isomorphism
invK : Br(K) Q/Z
for non-archimedean local fields K. If one uses the geometric Frobenius convention,
then by also adopting the crossed-product algebra method to define the isomor-
phism
Br(K)
H2(Ks/K,
Ks
×)
one would get the same composite isomorphism invK since the two sign differences
cancel out in the composite. (Beware that in [103] and [107] the Brauer group of a
general field K is defined to be
H2(Ks/K,
Ks
×),
and so the issue of choosing between
non-abelian cohomology or crossed-product algebras does not arise in the founda-
tional aspects of the theory. However, this issue implicitly arises in the relationship
of Brauer groups and central simple algebras, such as in [103, Appendix to §1]
where the details are omitted.)
Since Br(R) is cyclic of order 2 and Br(C) is trivial, for archimedean local fields
K there is a unique injective homomorphism invK : Br(K) Q/Z.
By [103, §1.1, Thm. 3], for a finite extension K /K of non-archimedean local
fields, composition with the natural map rK K : Br(K) Br(K ) satisfies
(1.2.4.1) invK rK
K
= [K : K] · invK .
By [107, Ch. XIII, §3, Cor. 3], invK (Δ) has order : K] for any central division
algebra Δ over K. These assertions are trivially verified to hold for archimedean
local fields K as well.
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