A Gentle Course in Local Class Field Theory: Local Number Fields, Brauer Groups, Galois Cohomology
Cambridge University Press, 12/20/2018
EAN 9781108421775, ISBN10: 1108421776
Hardcover, 306 pages, 25.4 x 17.8 x 2.5 cm
Language: English
Originally published in English
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.