Quadratic Residues and Non-Residues Selected Topics
This book offers an account of the classical theory of quadratic residues and non-residues with the goal of using that theory as a lens through which to view the development of some of the fundamental methods employed in modern elementary, algebraic, and
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Steve Wright
Quadratic Residues and Non-Residues Selected Topics
Lecture Notes in Mathematics Editors-in-Chief: J.-M. Morel, Cachan B. Teissier, Paris Advisory Board: Camillo De Lellis, Zurich Mario di Bernardo, Bristol Michel Brion, Grenoble Alessio Figalli, Zurich Davar Khoshnevisan, Salt Lake City Ioannis Kontoyiannis, Athens Gabor Lugosi, Barcelona Mark Podolskij, Aarhus Sylvia Serfaty, New York Anna Wienhard, Heidelberg
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More information about this series at http://www.springer.com/series/304
Steve Wright
Quadratic Residues and Non-Residues Selected Topics
123
Steve Wright Department of Mathematics and Statistics Oakland University Rochester Michigan, U.S.A.
ISSN 0075-8434 Lecture Notes in Mathematics ISBN 978-3-319-45954-7 DOI 10.1007/978-3-319-45955-4
ISSN 1617-9692 (electronic) ISBN 978-3-319-45955-4 (eBook)
Library of Congress Control Number: 2016956697 © Springer International Publishing Switzerland 2016 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 This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
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Preface
Although number theory as a coherent mathematical subject started with the work of Fermat in the 1630s, modern number theory, i.e., the systematic and mathematically rigorous development of the subject from fundamental properties of the integers, began in 1801 with the appearance of Gauss’ landmark treatise Disquisitiones Arithmeticae [19]. A major part of the Disquisitiones deals with quadratic residues and non-residues: if p is an odd prime, an integer z is a quadratic residue (respectively, a quadratic nonresidue) of p if there is (respectively, is not) an integer x such that x 2 ≡ z mod p. As we shall see, quadratic residues arise naturally as soon as one wants to solve the general quadratic congruence ax 2 + bx + c ≡ 0 mod m, a ≡ 0 mod m, and this, in fact
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