The Genus Fields of Algebraic Number Fields
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555 Makoto Ishida
The Genus Fields of Algebraic Number Fields
Springer-Verlag Berlin · Heidelberg· NewYork 1976
Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann
555 Makoto Ishida
The Genus Fields of Algebraic Number Fields
Springer-Verlag Berlin · Heidelberg· NewYork 1976
Author Makoto Ishida Department of Mathematics Tokyo Metropolitan University Fukazawa, Setagaya Tokyo 158/Japan
Library of Congress Cataloging in Publication Data
Isbida, Makoto, 1932The genus fields of algebraic nUlllber fields. (Lecture notes in mathematics; 555) Bibliogra:p!Jy: :p. Includes index. 1. Fields, Algebraic. 2. Class field theory. I. Title. II. Series: Lecture notes in mathematics (Berlin) ; 555. QA3.L28 no. 555 [QA247J 510' .Bs [512'.74J 76-49479
AMSSubjectClassifications(1970): 12A25, 12A30, 12A35, 12A40, 12A50, 12A65 ISBN 3-540-08000-7 SpringerNeriag Berlin' Heidelberg· New York ISBN 0-387-08000-7 Springer-Verlag New York' Heidelberg' Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin' Heidelberg 1976 Printed in Germany
These notes are taken from the lectures on algebraic number theory which I have given at several universities in Japan (Tokyo University of Education, Nagoya University, Kyushu University, Hokkaido University and Ochanomizu University)
and
include also the results obtained thereafter.
*
*
*
The genus theory (Theorie der Geschlechter) of quadratic number fields has its origin in 'Disquisitiones Arithmeticae' of Gauss
(cf. Cohn [3]).
In 1951, Hasse [12] gave a class field
theoretical interpretation of the theory. his introduction:
Here we quote from
'1st man einmal im Besitz der Hauptsgtze
der Klassenk5rpertheorie, so 19sst sich die Geschlechtertheorie ganz einfach in durchsichtiger, rein begrifflicher Gestalt herausarbeiten. ' Later on, the cyclic case was treated in Iyanaga and Tamagawa [22] and the abelian case in Leopoldt [23]. quote from the introduction in [23]
'Die Aufgabe der Theorie
ist es, aus den arithmetischen Eigenschaften von lechterk8rper g+
=
(K* : K)
K*
We also
K
der Gesch
explizit zu bestimmen, des sen Relativgrad die Anzahl der Geschlechter
im Mittelpunkt
der glteren Theorie stand.' Then in 1959, Fr5hlich
[6], [7] generalized the notion of
IV
the genus fields of algebraic number fields to not necessarily abelian case and obtained several interesting results on algebraic number fields of certain type.
After him, some contributions
have been made to the theory by Furuta [8J, Madan [24J, Frey and Geyer [5J and Ishida [15], *
*
[16],
[18],
[19],
[20J.
*
Chapter 1 is of preliminary nat
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