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Galois Cohomology
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Galois Cohomology
Suppose
is a finite Galois extension of fields, and
is a module for
. Put
Next suppose
is a module for the group
and for any extension
of
, let
all
We think of
as the group of elements of
that are ``defined over
''. For each
, put
Also, put
where
varies over all finite Galois extensions of
. (Recall: Galois means normal and separable.)
Example
11
.
4
.
1
The following are examples of
-modules:
where
is an elliptic curve over
.
Theorem
11
.
4
.
2
(Hilbert 90)
We have
.
Proof
. See [
Ser79
]. The main input to the proof is linear independence of automorphism and a clever little calculation.
William Stein 2012-09-24