The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local or global fields is an important tool in number theory.

The term Galois representation is frequently used when the $G$-module is a vector space over a field or a free module over a ring, but can also be used as a synonym for $G$-module. The study of Galois modules for extensions of local or global fields is an important tool in number theory.

Many objects that arise in number theory are naturally Galois representations. For example, if $L$ is a Galois extension of a number field $K$, the ring of integers $O_L$ of $L$ is a Galois module over OK for the Galois group of $L/K$ (see Hilbertâ€“Speiser theorem). If $K$ is a local field, the multiplicative group of its separable closure is a module for the absolute Galois group of $K$ and its study leads to local class field theory. For global class field theory, the union of the idele class groups of all finite separable extensions of $K$ is used instead.

There are also Galois representations that arise from auxiliary objects and can be used to study Galois groups. An important family of examples are the $\ell$-adic Tate modules of abelian varieties.

See also: Wikipedia article Galois module.