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

Front Cover
Cambridge University Press, 2018 - Mathematics - 306 pages
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.
 

Contents

Local number fields
23
Around Hensels lemma
31
Unramified extensions
43
Higher ramification groups
49
Infinite Galois extensions
62
The multiplicative structure of local number fields
72
BRAUER GROUPS
83
5
85
8
146
GALOIS COHOMOLOGY
157
10
187
Hilbert 90
204
Finer structure
214
CLASS FIELD THEORY
241
14
261
background material
281

Central simple algebras
107
7
122

Other editions - View all

Common terms and phrases

About the author (2018)

Pierre Guillot is a lecturer at the Université de Strasbourg and a researcher at the Institut de Recherche Mathématique Avancée (IRMA). He has authored numerous research papers in the areas of algebraic geometry, algebraic topology, quantum algebra, knot theory, combinatorics, the theory of Grothendieck's dessins d'enfants, and Galois cohomology.

Bibliographic information