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154 7. Galois Cohomology and Isomorphism Classification of Elliptic Curves

(5.2) Corollary (Hilbert’s “Theorem 90”). Let k /k be a finite cyclic extension

∗
with generator s of Gal(k /k). For x ∈ k the norm N k /k (x) = 1 if and only if there

exists y ∈ k with x = y/s(y). In other words, the following sequence is exact
∗
f N /k
k
∗
∗
∗
1 → k → k → k ∗ → k ,
where f (y) = y/s(y).
(5.3) Corollary. For a finite Galois extension k /k we have

H 1 Gal(k /k), SL n (k ) = 0.
induces the exact sequence
det 0 δ 1
0

∗

GL n (k) = H Gal, GL n (k ) −→ k = H (Gal, GL 1 ) → H (Gal, SL n ) → 1.
1
Since det:GL n (k) → k is surjective and H (Gal, GL n ) = 0, we deduce that
∗
1
H (Gal(k /k), SL n (k )) = 1.

Let µ n (k) denote the nth roots of unity contained in k , and observe that µ n (k )
∗

is a Gal(k /k) submodule of k ∗ for any Galois extension k /k.For n prime to the

characteristic of k and k separably algebraically closed, the following sequence is

exact
n

∗
∗
1 → µ n (k ) → k → k → 1.
Applying the exact cohomology sequence, using (4.5), and using
0
∗

∗
H (Gal(k /k), k ) = k ,
we have the next proposition.

(5.4) Proposition (Kummer Sequence). For k a separably algebraically closed
Galois extension of k and n prime to the characteristic of k the sequence
n 1

∗
∗

1 → µ n (k) → k → k → H (Gal(k /k), µ n (k /k)) → 1
is exact, and we have an isomorphism
∗ n
1
∗

k /(k ) → H (Gal(k /k), µ n (k )).
Further, if k contains the nth roots of unity, then
1
∗ n

∗
k /(k ) → H (Gal(k /k), µ n (k )) = Hom(Gal(k /k), µ n (k )).
is an isomorphism.
∗ 2
For example, in characteristic different from 2, the nonzero elements of k /(k )
∗

correspond to the subgroups of index 2 in Gal(k /k), that is to quadratic extensions
of k.