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 [Δ : K] for any central division

algebra Δ over K. These assertions are trivially verified to hold for archimedean

local fields K as well.