INTRODUCTION

Let N/K be a finite Galois extension of number fields and let Γ = Gal(N/K) denote its Galois group. In these lectures we study the structures of certain ‘Galois modules’ arising in this context such as the ring of algebraic integers ON of N, the multiplicative group UN of units of ON modulo torsion elements, and the ideal class group ClN of N. A striking feature of the theory discussed is the close interplay between the module structure and the arithmetic of the number fields. In particular (Artin) L-functions play a fundamental rôle in various places. What results however is much more than theorems and conjectures relating two very abstract and sophisticated points of view – the module theoretic aspect is frequently very down to earth, and indeed some indication of the power of the theory lies in its capacity to provide concrete and explicit information.

There are several approaches to these Galois module theoretic problems and we shall discuss different approaches in separate paragraphs.

The case which is best understood is that of ON when N/K is at most tamely ramified. Here, ON is a (locally-free) Z[Γ]-module and in §1 we discuss the fundamental result which shows that its structure as such is determined, up to stable isomorphism, by the root numbers occuring in the functional equation of the Artin L-functions of N/K attached to the irreducible symplectic characters of Γ.