5.5. Hilbert’s Theorem 90.

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Let G be a finite group. A G-module is an abelian group M together with an action of G,

i.e., a map G ×M → M such that

(a) σ(m+ m_) = σm + σm_ for all σ ∈ G, m,m_ ∈ M;

(b) (στ)(m) = σ(τm) for all σ, τ ∈ G, m ∈ M;

(c) 1m = m for all m ∈ M.

Thus, to give an action of G on M is the same as to give a homomorphism G → Aut(M)

(automorphisms of M as an abelian group).

Example 5.15. Let E be a Galois extension of F, with Galois group G;the n (E, +) and

E× are G-modules.

Let M be a G-module. A crossed homomorphism is a map f : G → M such that

f(στ) = f(σ) + σf(τ ).

Note that the condition implies that f(1) = f(1 ・ 1) = f(1) + f(1), and so f(1) = 0.

Example 5.16. (a) Consider a crossed homomorphism f : G → M, and let σ ∈ G. Then

f(σ2) = f(σ) + σf(σ),

f(σ3) = f(σ ・ σ2) = f(σ) + σf(σ) + σ2f(σ)

and so on, until

f(σn) = f(σ) + σf(σ) + ・ ・ ・ + σn−1f(σ).

Thus, if G is a cyclic group of order n generated by σ, then a crossed homomorphism

f : G → M is determined by f(σ) = x, and x satisfies the equation

x + σx + ・ ・ ・ + σn−1x = 0, (∗)

Conversely, if x ∈ M satisfies (*), then the formulas f(σi) = x + σx + ・ ・ ・ + σi−1x define a

crossed homomorphism f : G → M. In this case we have a one-to-one correspondence

{crossed homs f : G → M} f_→↔f(σ) {x ∈ M satisfying (∗)}.

(b) For any x ∈ M, we obtain a crossed homomorphism by putting

f(σ) = σx − x, all σ ∈ G.

Such a crossed homomorphism is called a principal crossed homomorphism.

(c) If G acts trivially on M, i.e., σm = m for all σ ∈ G and m ∈ M, then a crossed

homomorphism is simply a homomorphism, and there are no nontrivial principal crossed

homomorphisms.

The sum of two crossed homomorphisms is again a crossed homomorphism, and the sum

of two principal crossed homomorphisms is again principal. Thus we can define

H1(G,M) =

{crossed homomorphisms}

{principal crossed homomorphisms}.

FIELDS AND GALOIS THEORY 43

The cohomology groups Hn(G,M) have been defined for all n ∈ N, but since this was not

done until the twentieth century, it will not be discussed in this course.

Example 5.17. Let π : _X → X be the universal covering space of a topological space X,

and let Γ be the group of covering transformations. Under some fairly general hypotheses, a

Γ-module M will define a sheaf M on X, and H1(X,M) ≈ H1(Γ,M). For example, when

M = Z with the trivial action of Γ, this becomes the isomorphism H1(X, Z) ≈ H1(Γ, Z) =

Hom(Γ, Z).

Theorem 5.18. Let E be a Galois extension of F with group G; then H1(G,E×) = 0,

i.e., every crossed homomorphism G → E× is principal.

Proof. Let f be a crossed homomorphism G → E×. In multiplicative notation, this

means,

f(στ) = f(σ) ・ σ(f(τ )), σ,τ∈ G,

and we have to find a γ ∈ E× such that f(σ) = σγ/γ for all σ ∈ G. Because the f(τ) are

nonzero, Dedekind’s theorem implies that

_

f(τ )τ : E → E

is not the zero map, i.e., there exists an α ∈ E such that

β =

_

τ∈G

f(τ )τα _= 0.

But then, for σ ∈ G,

σβ =

_

τ∈G

σ(f(τ )) ・ στ(α) =

_

τ∈G

f(σ)

−1 f(στ) ・ στ(α) = f(σ)

−1

_

τ∈G

f(στ)στ(α) = f(σ)

−1β,

which shows that f(σ) = β

σ(β) and so we can take β = γ−1.

Let E be a Galois extension of F with Galois group G. We define the norm of an element

α ∈ E to be

Nmα =

_

σ∈G

σα.

Then, for τ ∈ G,

τ(Nmα) =

_

σ∈G

τσα = Nmα,

and so Nmα ∈ F. The map α → Nmα : E× → F× is a homomorphism. For example, the

norm map C× → R× is α → |α|2 and the norm map Q[

√

d]× → Q× is a + b

√

d → a2 − db2.

We are interested in determining the kernel of this homomorphism. Clearly if α is of the

form β

τβ, then Nm(α) = 1. Our next result show that, for cyclic extensions, all elements with

norm 1 are of this form.

Corollary 5.19 (Hilbert’s theorem 90). 7Let E be a finite cyclic extension of F with

Galois group < σ >; if NmE/F α = 1, then α = β/σβ for some β ∈ E.

7The theorem is Satz 90 in Hilbert’s book, Theorie der Algebraische ZahlkЁorper, 1897, which laid the

foundations for modern algebraic number theory. Many point to it as a book that made a fundamental

contribution to mathematical progress, but Emil Artin has been quoted as saying that it set number theory

back thirty years—it wasn’t sufficiently abstract for his taste.

44 J.S. MILNE

Proof. Let m = [E : F]. The condition on α is that α ・ σα ・ ・ ・ σm−1α = 1, and so (see

5.16a) there is a crossed homomorphism f :<σ>→ E× with f(σ) = α. The theorem now

shows that f is principal, which means that there is a β with f(σ) = β/σβ.

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