The Quadratic Reciprocity Law A Collection of Classical Proofs
This book is the English translation of Baumgart’s thesis on the early proofs of the quadratic reciprocity law (“Über das quadratische Reciprocitätsgesetz. Eine vergleichende Darstellung der Beweise”), first published in 1885. It is divided into two parts
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The Quadratic Reciprocity Law A Collection of Classical Proofs Edited and Translated by Franz Lemmermeyer
The Quadratic Reciprocity Law
Oswald Baumgart
The Quadratic Reciprocity Law A Collection of Classical Proofs Edited and Translated by Franz Lemmermeyer
Oswald Baumgart Translated by Franz Lemmermeyer Jagstzell, Germany
ISBN 978-3-319-16282-9 DOI 10.1007/978-3-319-16283-6
ISBN 978-3-319-16283-6 (eBook)
Library of Congress Control Number: 2015938466 Mathematics Subject Classification (2010): 11A15, 11T22, 11T24 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG and Switzerland is part of Springer Science+Business Media (www.birkhauser-science.com)
Translator’s Preface
This book contains a translation of Baumgart’s thesis Über das quadratische Reciprocitätsgesetz. Eine vergleichende Darstellung der Beweise (On the Quadratic Reciprocity Law. A Comparative Presentation of its Proofs) written in Göttingen a hundred years after Legendre’s publication of the reciprocity law. Its aim was to present and compare the proofs of the quadratic reciprocity law known to him in 1885. The introduction to this book, which was to give a detailed history of the number theory leading up to the statement and the first proofs of the quadratic reciprocity law, eventually evolved into a book of its own and will be published separately. Baumgart’s thesis consists of two parts. The first part gives a very brief history of the development up to Legendre and then presents some different proofs of the quadratic reciprocity law; Baumgart distinguished Gauss’s first proof by induction, proofs by Gauss’s Lemma, by complex analysis, by cyclotomy, and by quadratic forms. The last two chapters deal with supplementary laws and algorithms for computing Legendre symbols. In the second part, Baumgart compares the principles of these proo
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