A Gentle Course in Local Class Field Theory
Local Number Fields, Brauer Groups, Galois Cohomology

A self-contained exposition of local class field theory for students in advanced algebra.

Pierre Guillot (Author)

9781108432245, Cambridge University Press

Paperback / softback, published 1 November 2018

306 pages
24.7 x 17.4 x 1.6 cm, 0.53 kg

'... a valuable book. I certainly cannot think of any other source that makes the basic ideas of class field theory, and the Kronecker-Weber theorems, more accessible. And the background material on noncommutative algebra and group cohomology can be read with profit by somebody just interested in these topics alone. Highly recommended.' Mark Hunacek, The Mathematical Gazette

This book offers a self-contained exposition of local class field theory, serving as a second course on Galois theory. It opens with a discussion of several fundamental topics in algebra, such as profinite groups, p-adic fields, semisimple algebras and their modules, and homological algebra with the example of group cohomology. The book culminates with the description of the abelian extensions of local number fields, as well as the celebrated Kronecker–Weber theory, in both the local and global cases. The material will find use across disciplines, including number theory, representation theory, algebraic geometry, and algebraic topology. Written for beginning graduate students and advanced undergraduates, this book can be used in the classroom or for independent study.

Part I. Preliminaries: 1. Kummer theory
2. Local number fields
3. Tools from topology
4. The multiplicative structure of local number fields
Part II. Brauer Groups: 5. Skewfields, algebras, and modules
6. Central simple algebras
7. Combinatorial constructions
8. The Brauer group of a local number field
Part III. Galois Cohomology: 9. Ext and Tor
10. Group cohomology
11. Hilbert 90
12. Finer structure
Part IV. Class Field Theory: 13. Local class field theory
14. An introduction to number fields.

Subject Areas: Algebraic topology [PBPD], Algebraic geometry [PBMW], Number theory [PBH], Algebra [PBF]