Galois representation

In mathematics, a Galois module is a G-module, with G being the Galois group 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, 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.

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.

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