A Galois representation of a field 
on a finite-dimensional vector
space 
over a field 
is a group homomorphism
 |
where 
is the Galois group of an algebraic
closure 
of 
and 
is the general linear group of 
. When the groups carry natural topologies, a Galois representation
is normally required to be continuous (Serre 1998).
Galois representations translate arithmetic information into linear algebra. For example, the action of 
on the torsion points of an elliptic
curve over 
gives finite and 
-adic
Galois representations. Such representations connect elliptic curves with modular
forms and are central to the proof of Fermat's
last theorem.
