Suppose is a finite Galois extension of
fields, and is a module for
Next suppose is a module for the group
and for any extension of , let
We think of as the group of elements of that are
``defined over ''.
For each , put
where varies over all finite Galois extensions of .
(Recall: Galois means normal and separable.)
The following are examples of
is an elliptic curve over
]. The main input to the proof is linear
independence of automorphism and a clever little calculation.