Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.
Concept
Galois module
Summary
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 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.
Given a field K, the multiplicative group (Ks)× of a separable closure of K is a Galois module for the absolute Galois group. Its second cohomology group is isomorphic to the Brauer group of K (by Hilbert's theorem 90, its first cohomology group is zero).
If X is a smooth proper scheme over a field K then the l-adic cohomology groups of its geometric fibre are Galois modules for the absolute Galois group of K.
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 Iw denote its inertia group. A Galois module ρ : G → Aut(V) is said to be unramified if ρ(Iw) = {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 OL of algebraic integers of L can be considered as an OK[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 OL such that its conjugate elements under G give a free basis for OL over OK. This is an interesting question even (perhaps especially) when K is the rational number field Q.
For example, if L = Q(), is there a normal integral basis? The answer is yes, as one sees by identifying it with Q(ζ) where
ζ = exp(2pii/3).
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).
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
Related publications (39)