Octic Reciprocity
The proof of the octic reciprocity law differs considerably from those in the cubic and quartic cases; in fact one needs methods that are much more sophisticated than Gauss sums, and this is why we resurrect elliptic Gauss sums These were introduced by Ei
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Springer-Verlag Berlin Heidelberg GmbH
Franz Lemmermeyer
Reciprocity laws From Euler to Eisenstein
'Spring er
Franz Lemmermeyer
http:/ /www.rzuser. uni-heidelberg.de/-hb3/ e-mail: [email protected]
Library of Congress Cataloging-in-Publication Data Lemmermeyer, Franz, 1962Reciprocity laws: from Euler to Eisenstein I Franz Lemmer meyer. p. em.-- (Springer monographs in mathematics) Includes bibliographical references and index. ISBN 978-3-662-12893-0 (eBook) ISBN 978-3-642-08628-1 DOI 10.1007/978-3-662-12893-0 1. Reciprocity theorems I. Title II. Series QA24l.L562000 512'74--dc2I
00-026905
Mathematics Subject Classification (1991): llA15
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Preface
The history of reciprocity laws is a history of algebraic number theory. This is a book on reciprocity laws, and our introductory remark is placed at the beginning as a warning: in fact a reader who is acquainted with little more than a course in elementary number theory may be surprised to learn that quadratic reciprocity does - in a sense that we will explain - belong to the realm of algebraic number theory. Heeke {[348, p. 59]) has formulated this as follows: Von der Entdeckung des Reziprozitiitsgesetzes kann man die moderne Zahlentheorie datieren. Seiner Form nach gehort es noch der Theorie der rationalen Zahlen an, es liillt sich aussprechen als eine Beziehung lediglich zwischen rationalen Zahlen; jedoch weist es seinem lnhalt nach iiber den Bereich der rationalen Zahlen hinaus. [... ] Die Entwicklung der algebraischen Zahlentheorie hat nun wirklich gezeigt, daB der lnhalt des quadratischen Reziprozitiitsgesetzes erst verstiindlich wird, wenn man zu den allgemeinen algebraischen Zahlen iibergeht, und daB ein dem Wesen des Problems angemessener Beweis sich am besten mit diesen hoheren Hilfsmitteln fiihren liiBt, wiihrend man von den elementaren Beweisen
