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Galois Module Structure
Victor P. Snaith, McMaster University, Hamilton, ON, Canada
A co-publication of the AMS and Fields Institute.
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Fields Institute Monographs
1994; 207 pp; hardcover
Volume: 2
ISBN-10: 0-8218-0264-X
ISBN-13: 978-0-8218-0264-9
List Price: US$47
Member Price: US$37.60
Order Code: FIM/2
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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 idèlic-valued functions on the complex representations of \(G\). Following a theme pioneered by A. Frölich, 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 \(s = -1\).

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

Readership

Graduate students in number theory and more postdoctoral research mathematicians in number theory and algebra.

Reviews

"This is an advanced text ... However, the author does make plain what he requires and gives sources. He also provides a good introduction to the book as a whole and to its various topics, so that the reader knows the line of argument. There are many explicit examples, and exercises are provided for each chapter ... The book will also be invaluable as introductory reading for a student working in isolation, and it should find its way to the bookshelves of all who work with Galois modules."

-- Mathematical Reviews

"It will be useful for specialists inside and outside of this subject, as well as for graduate students."

-- Zentralblatt MATH

"Should be useful to mathematicians who are interested in learning about certain aspects of the theory of arithmetic Galois module structure, as well as to specialists in the area."

-- Bulletin of the AMS

Table of Contents

  • Basic preliminaries
  • Class-groups
  • Logarithmic techniques
  • The tame case
  • Maximal orders
  • Quaternionic examples
  • Higher algebraic \(K\)-theory
  • Index
  • Bibliography
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