Galois Module - Galois Representations in Number Theory

Galois Representations 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 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.