Multiplicative Galois Module Structure

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.