Galois cohomology is the continuous cohomology of Galois groups. More precisely, let 
be a Galois extension
field of 
with Galois group 
, and let 
be a discrete 
-module on which 
acts continuously. The Galois cohomology groups of 
with coefficients in 
are
 |
For a separable closure 
of 
, the usual absolute notation
is
 |
In particular, 
is the subgroup of elements fixed by the absolute Galois
group (Serre 1997).
For an algebraic group 
over 
, the first Galois cohomology
is generally a pointed set rather than a group. Its elements classify the isomorphism
classes of principal homogeneous spaces
under 
. When 
is commutative, 
is an abelian group.
For an elliptic curve 
, this gives the group 
used in the definition of the Tate-Shafarevich
group.
