Class Field Theory

-The Bonn Lectures- Edited by Alexander Schmidt

verfasst von: Jürgen Neukirch

Verlag: Springer Berlin Heidelberg

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

The present manuscript is an improved edition of a text that first appeared under the same title in Bonner Mathematische Schriften, no.26, and originated from a series of lectures given by the author in 1965/66 in Wolfgang Krull's seminar in Bonn. Its main goal is to provide the reader, acquainted with the basics of algebraic number theory, a quick and immediate access to class field theory. This script consists of three parts, the first of which discusses the cohomology of finite groups. The second part discusses local class field theory, and the third part concerns the class field theory of finite algebraic number fields.

Inhaltsverzeichnis

Frontmatter

Part I Cohomology of Finite Groups

Abstract

The cohomology of finite groups deals with a general situation that occurs frequently in different concrete forms. For example, if L|K is a finite Galois extension with Galois group G, then G acts on the multiplicative group L ^× of the extension field L. In the special case of an extension of finite algebraic number fields, G acts on the ideal group J of the extension field L. The theory of group extensions provides us with the following example: If G is an abstract finite group and A is a normal subgroup, then G acts on A via conjugation. In representation theory we study matrix groups G that act on a vector space. The basic notion underlying all these examples is that of a G-module. We will now present some general considerations about G-modules, some of which the reader may already know from the theory of modules over general rings.

Jürgen Neukirch

Part II Local Class Field Theory

Abstract

Local and global class field theory, as well as a series of further theories for which the name class field theory is similarly justified, have the following principle in common. All of these theories involve a canonical bijective correspondence between the abelian extensions of a field K and certain subgroups of a corresponding module AK associated with the field K.

Jürgen Neukirch

Part III Global Class Field Theory

Abstract

We assume the reader is familiar with the basic concepts and theorems of algebraic number theory for which we refer to the standard text books, for example, [6], [21], [30]. Nevertheless, in this section we briefly summarize the for us most important facts.

Jürgen Neukirch

Backmatter

Titel: Class Field Theory
verfasst von: Jürgen Neukirch
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-642-35437-3
Print ISBN: 978-3-642-35436-6
DOI: https://doi.org/10.1007/978-3-642-35437-3