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Full Description
The book contains the main results of class field theory and Artin L functions, both for number fields and function fields, together with the necessary foundations concerning topological groups, cohomology, and simple algebras.
While the first three chapters presuppose only basic algebraic and topological knowledge, the rest of the books assumes knowledge of the basic theory of algebraic numbers and algebraic functions, such as those contained in my previous book, An Invitation to Algebraic Numbers and Algebraic Functions (CRC Press, 2020).
The main features of the book are:
A detailed study of Pontrjagin's dualtiy theorem.
A thorough presentation of the cohomology of profinite groups.
A introduction to simple algebras.
An extensive discussion of the various ray class groups, both in the divisor-theoretic and the idelic language.
The presentation of local and global class field theory in the algebra-theoretic concept of H. Hasse.
The study of holomorphy domains and their relevance for class field theory.
Simple classical proofs of the functional equation for L functions both for number fields and function fields.
A self-contained presentation of the theorems of representation theory needed for Artin L functions.
Application of Artin L functions for arithmetical results.
Contents
Topological groups and infinite Galois theory
Cohomology of groups
Simple algebras
Local class field theory
Global fields: Adeles, ideles and holomorphy domains
Global class field theory
Functional equations and Artin L functions
