Galois Module Structure

Galois module structure deals with the construction of algebraic invariants from a Galois extension of number fields with group $G$. Typically these invariants lie in the class-group of some group-ring of $G$ or of a related order. These class-groups have ``Hom-descriptions'' in terms of idelic-valued functions on the complex representations of $G$. Following a theme pioneered by A. Frolich, T. Chinburg constructed several invariants whose Hom-descriptions are (conjecturally) given in terms of Artin root numbers. For a tame extension, the second Chinburg invariant is given by the ring of integers, and M. J. Taylor proved the conjecture in this case. The first published graduate course on the Chinburg conjectures, this book provides the necessary background in algebraic and analytic number theory, cohomology, representation theory, and Hom-descriptions. The computation of Hom-descriptions is facilitated by Snaith's Explicit Brauer Induction technique in representation theory. In this way, illustrative special cases of the main results and new examples of the conjectures are proved and amplified by numerous exercises and research problems. The final chapter introduces a new invariant constructed from algebraic $K$-theory, whose Hom-description is related to the $L$-function value at $s = -1$.