Galois module

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In mathematics, and in particular in algebraic number theory, a Galois module is a module for a Galois group G. Equivalently, for a Galois group G and a group ring R[G] of G with respect to some ring R, a Galois module is some R[G]-module M. In that general sense the term Galois representation is a synonym. There are a number of important, much more specialized cases. Certainly G can either be an infinite profinite Galois group, or a finite one for a finite Galois extension L/K of fields.

In the case of G profinite, there is a large supply of G-modules available in the theory of étale cohomology, which is an algebraic theory (and therefore exhibits 'covariance' with respect to Galois symmetry). A basic discovery of the 1960s is that such modules are as non-trivial as they can be, in general, so that the theory is rather rich.

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, so of α in O_K 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

ζ = exp(2πi/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). 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.

Further examples of Galois module theories are those of the Galois action on the unit group of O_L, and on the ideal class group (leading to Iwasawa theory).