Basic Algebraic Structures and Elementary Number Theory
In this preliminary chapter, we lay the algebraic foundations which we believe are necessary to understand the theory of finite fields and their applications. Although most of this chapter is certainly covered by many text books on Algebra, we decided to
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Dirk Hachenberger Dieter Jungnickel
Topics in Galois Fields
Algorithms and Computation in Mathematics Volume 29 Series Editors David Eisenbud, Berkeley, CA, USA Michael F. Singer, Department of Mathematics, Raleigh, NC, USA Bernd Sturmfels, Berkeley, CA, USA Mark Braverman, Princeton, NJ, USA Bianca Viray, Department of Mathematics, University of Washington, Seattle, WA, USA
More information about this series at http://www.springer.com/series/3339
Dirk Hachenberger Dieter Jungnickel •
Topics in Galois Fields
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Dirk Hachenberger Department of Mathematics University of Augsburg Augsburg, Germany
Dieter Jungnickel Department of Mathematics University of Augsburg Augsburg, Germany
ISSN 1431-1550 Algorithms and Computation in Mathematics ISBN 978-3-030-60804-0 ISBN 978-3-030-60806-4 https://doi.org/10.1007/978-3-030-60806-4
(eBook)
Mathematics Subject Classification: 11TXX, 12E20, 15B33, 94A55 © Springer Nature Switzerland AG 2020 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, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
To my parents, Carola and Rudi D IRK H ACHENBERGER
To the memory of my mother D IETER J UNGNICKEL
Preface
The idea of writing a book Topics in Galois Fields originated a long time ago, maybe in 1999, when the Fifth International Conference on Finite Fields and Applications was hosted by our home university in Augsburg, Germany. Although we could draw on our two previous monographs, Finite Fields – Structure and Arithmetics [211] by Dieter Jungnickel and Finite Fields – Normal Bases and Completely Free Elements [161] by Dirk Hachenberger, this proved to be a decidedly long-term project, which of course had to do with numerous other duties that had to be fulfilled. We are therefore very grateful to our publ
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