Survey of Results

Abstract

The questions covered by the heading “Galois module structure of algebraic integers” are quite classical in their origin. The power, depth and interest of the theory however stem from its recently discovered connection with arithmetic invariants, associated with representations of Galois groups, in particular those coming from the functional equation of the Artin L-function. Since this discovery, the subject which previously had lain rather barren, has developed rapidly. It now has a rich and powerful body of theorems, which can effectively be used to provide quite explicit, concrete information, and which at the same time also lead to new insight into other branches of algebraic number theory, such as the Stickelberger relations, the Galois module structure of ideal classgroups, and the embedding problem. In particular one must mention here the study of the arithmetic nature of local and global root numbers and Galois Gauss sums, and indeed this can be taken as a second main topic of the present volume. The aim of the present section is to outline in approximate — but not strict — chronological order the several strands of research and their interaction, which led to the development of the theory in the first place.