Galois Theory
The first chapter is an exposition of Galois theory and its applications to the questions of solvability of algebraic equations in explicit form. Apart from the classical problem on solvability of an algebraic equation by radicals, we also consider other
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Askold Khovanskii
Galois Theory, Coverings, and Riemann Surfaces
Askold Khovanskii Dept. Mathematics University of Toronto Toronto, Ontario, Canada
ISBN 978-3-642-38840-8 ISBN 978-3-642-38841-5 (eBook) DOI 10.1007/978-3-642-38841-5 Springer Heidelberg New York Dordrecht London Library of Congress Control Number: 2013949677 Mathematics Subject Classification (2010): 55-02, 12F10, 30F10 © Springer-Verlag Berlin Heidelberg 2013 Translation of Russian edition entitled “Teoriya Galua, Nakrytiya i Rimanovy Poverkhnosti”, published by MCCME, Moscow, Russia, 2006 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. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. 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. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
Preface
The main goal of this book is an exposition of Galois theory and its applications to the questions of solvability of algebraic equations in explicit form. Apart from the classical problem on solvability of an algebraic equation by radicals, we also consider other problems of this type, for instance, the question of solvability of an equation by radicals and by solving auxiliary equations of degree at most k. There exists a surprising analogy between the fundamental theorem of Galois theory and classification of coverings over a topological space. A description of this analogy is the second goal of the present book. We consider several classifications o
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