Class Field Theory: -The Bonn Lectures- Edited by Alexander Schmidt

195 Pages·2013·1.66 MB·English

by  Jürgen Neukirch (auth.)

Jürgen Neukirch
Class Field
Theory
– The Bonn Lectures –
Edited by Alexander Schmidt
Class Field Theory

Jürgen Neukirch
Class Field Theory
-The Bonn Lectures-
Edited by Alexander Schmidt
Translated from the German
by F. Lemmermeyer and W. Snyder
Language Editor: A. Rosenschon
Jürgen Neukirch †
Editor
Alexander Schmidt
Mathematical Institute
Heidelberg University
Germany

ISBN 978-3-642-35436-6 ISBN 978-3-642-35437-3 (eBook)
DOI 10.1007/978-3-642-35437-3
Springer Heidelberg New York Dordrecht London

Library of Congress Control Number: 2013935211

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Foreword
In 1969, Ju¨rgen Neukirch’s book Klassenko¨rpertheorie was published by Bib-
liographisches Institut Mannheim. The main goal of the book was to grant
the reader, who has acquainted himself with the basics of algebraic number
theory, a quick and immediate access to class field theory.
Although this book has been out of print for many years, it has remained
the favorite introduction to class field theory in Germany. As a student in
the 1980s, I myself studied a copy from the library that showed clear signs of
extensive use. This motivated the idea to make the text available again, as a
printed book as well as a freely accessible file for downloading.
ThisbookshouldnotbeconfusedwithNeukirch’sbook“ClassFieldTheory”
(SpringerGrundlehrenvol.280,1986),whichhasanotherfocus.Thetextpre-
sented here is essentially identical with the German text based on Neukirch’s
original Bonn lectures; I only corrected mistakes and updated notation.
I would like to thank Rita Neukirch for her generous support for this new
English edition of the work of her late husband. I also thank Rosina Bonn
for her excellent typesetting of the original German text into LATEX and my
friends and colleagues Andreas Rosenschon, Bernd Schober and Malte Witte
for their invaluable help in improving the English edition.
Heidelberg, October 2012 Alexander Schmidt

Dedicated to my teacher
WOLFGANG KRULL

Preface
Thepresentmanuscriptisanimprovededitionofatextthatfirstappearedun-
der the same title in Bonner Mathematische Schriften, no.26, and originated
from a series of lectures given by the author in 1965/66 in W. Krull’s semi-
nar in Bonn. Since the mathematical literature lacked a uniform presentation
of class field theory based on modern cohomological methods, a summarizing
expositionoftheselecturesseemedtobeuseful.Themaingoalwastoprovide
the reader, who has acquainted himself with the basics of algebraic number
theory, a quick and immediate access to class field theory.
Thisscriptconsistsofthreeparts,thefirstofwhichdiscussesthecohomology
offinitegroups.Nowadays,cohomologyhasconqueredlargeareasofalgebraic
number theory. Nevertheless, the question whether class field theory can be
done without this machinery is a frequent topic of discussion. However, apart
from the possibility of formulating the theory in terms of algebras, which is
closely related to cohomology, we do not dispose of such a theory at this
point, although recent results due to J. Lubin and J. Tate on the explicit
determinationofthelocalnormresiduesymbolprovidesomesupportforthis
viewpoint. But one must not overlook the fact that cohomology presents –
in particular for the learner – a wealth of far reaching advantages. In class
field theory, cohomology plays the role of a calculus that allows a clear and
logical development of the theory under a unified viewpoint. Its importance,
however,isbynomeans onlyof formal nature.In fact,localclass fieldtheory
could originally be developed by defining the norm residue symbol via the
Frobenius automorphism for unramified extensions only. It was cohomology
that gave a vital impetus to the theory, making also the ramified extensions
accessibletoclassfieldtheoreticmethods.Thisrelationshipwasdiscoveredby
H. Hasse and had an immediate impact also on the global theory. Although
it was formulated in the language of algebras at first, the cohomological prin-
ciples behind it did not remain hidden for long. In addition, beyond class
field theory, the use of cohomological methods in general field theory has led,
via Galois cohomology, to a wealth of far reaching results with a novel allure.