This book is the result of a short course on the Galois structure of $S$-units that was given at The Fields Institute in the fall of 1993. Offering a new angle on an old problem, the main theme is that this structure should be determined by class field theory, in its cohomological form, and by the behavior of Artin $L$-functions at $s=0$. A proof of this--or even a precise formulation--is still far away, but the available evidence all points in this direction. The work brings together the current evidence that the Galois structure of $S$-units can be described.

Titles in this series are co-published with The Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).

Readership

Graduate students and research mathematicians, specifically algebraic number theorists.

Reviews

"No comparable work exists in the literature ... we should be thankful to have this book, which is at the same time an introduction, and a report on the state of the art, written by one of the leading experts ... Even non-experts with some background in algebra should be able to profit from browsing in this book, which will be a standard reference for a long time to come."

-- Bulletin of the AMS

Table of Contents

• Overview
• From class field theory
• Extension classes
• Locally free class groups
• Tate sequences
• Recognizing $G$-modules
• Local analogue
• $\Omega _m$ and the $G$-module structure of $E$
• Artin $L$-functions at $s=0$
• $q$-indices
• Parallel properties of$A_\varphi$ and $A_\varphi$
• $\mathbb Q$-valued characters
• Representing the Chinburg class
• Small $S$
• A cyclotomic example
• Notes
• Bibliography
• Subject index