Galois modules and class field theory Boas

10. Galois modules and class field theory Boas Erez In this section we shall try to present the reader with a sample of several significant instances where, on the way to proving results in Galois module theory, one is lead to use class field theory. Conversely, some contributions of Galois module theory to class fields theory are hinted at. We shall also single out some problems that in our opinion deserve further attention. The Normal Basis Theorem is one of the basic results in the Galois theory of fields. In fact one can use it to obtain a proof of the fundamental theorem of the theory, which sets up a correspondence between subgroups of the Galois group and subfields. Let us recall its statement and give a version of its proof following E. Noether and M. Deuring (a very modern proof!). Theorem (Noether, Deuring). Let K be a finite extension of Q. Let L/K be a finite Galois extension with Galois group G = Gal(L/K). Then L is isomorphic to K[G] as a K[G]-module. That is: there is an a ∈ L such that {σ(a)} σ∈G is a K-basis of L. Such an a is called a normal basis generator of L over K. then apply the Krull–Schmidt theorem to deduce that this isomorphism descends to K. Note that an element a in L generates a normal basis of L over K if and only if ϕ(a) ∈ L[G] * .