In mathematics, a Galois module is a G-module, with G being the Galois group (named for Évariste Galois) of some extension of fields. 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 in representation theory, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local or global fields and their group cohomology is an important tool in number theory.

Let K be a valued field (with valuation denoted v) and let L/K be a finite Galois extension with Galois group G. For an extension w of v to L, let I_w denote its inertia group. A Galois module ρ : G → Aut(V) is said to be unramified if ρ(I_w) = {1}.

In classical algebraic number theory, let L be a Galois extension of a field K, and let G be the corresponding Galois group. Then the ring O_L of algebraic integers of L can be considered as an O_K[G]-module, and one can ask what its structure is. This is an arithmetic question, in that by the normal basis theorem one knows that L is a free K[G]-module of rank 1. If the same is true for the integers, that is equivalent to the existence of a normal integral basis, i.e. of α in O_L such that its conjugate elements under G give a free basis for O_L over O_K. This is an interesting question even (perhaps especially) when K is the rational number field Q.

For example, if L = Q(√−3), is there a normal integral basis? The answer is yes, as one sees by identifying it with Q(ζ) where

In fact all the subfields of the cyclotomic fields for p-th roots of unity for p a prime number have normal integral bases (over Z), as can be deduced from the theory of Gaussian periods (the Hilbert–Speiser theorem). On the other hand, the Gaussian field does not. This is an example of a necessary condition found by Emmy Noether (perhaps known earlier?). What matters here is tame ramification. In terms of the discriminant D of L, and taking still K = Q, no prime p must divide D to the power p. Then Noether's theorem states that tame ramification is necessary and sufficient for O_L to be a projective module over Z[G]. It is certainly therefore necessary for it to be a free module. It leaves the question of the gap between free and projective, for which a large theory has now been built up.

A classical result, based on a result of David Hilbert, is that a tamely ramified abelian number field has a normal integral basis. This may be seen by using the Kronecker–Weber theorem to embed the abelian field into a cyclotomic field.

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 O_K for the Galois group of L/K (see Hilbert–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 ℓ-adic Tate modules of abelian varieties.