A Gentle Course in Local Class Field Theory Local Number Fields Brauer Groups Galois Cohomology 1st Edition Pierre Guillot 1108389864 9781108389860

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Product details:

• ISBN 10: 1108389864
• ISBN 13: 9781108389860
• Author: Pierre Guillot

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.

Table of contents:

1 Kummer theory

2 Local number fields

3 Tools from topology

4 The multiplicative structure of local number fields

5 Skewfields, algebras, and modules

6 Central simple algebras

7 Combinatorial constructions

8 The Brauer group of a local number field

9 Ext and Tor

10 Group cohomology

11 Hilbert 90

12 Finer structure

13 Local class field theory

14 An introduction to number fields

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