Galois Theory, Second Edition PDF
Preview Galois Theory, Second Edition
r Monographs Theory Galois Second Edition S David A. Cox GALOIS THEORY PURE AND APPLIED MATHEMATICS A Wiley Series of Texts, Monographs, and Tracts Founded by RICHARD COURANT Editors Emeriti: MYRON B. ALLEN III, DAVID A. COX, PETER HILTON, HARRY HOCHSTADT, PETER LAX, JOHN TOLAND A complete list of the titles in this series appears at the end of this volume. GALOIS THEORY Second Edition David A. Cox Department of Mathematics Amherst College Amherst, MA WILEY & SONS, INC., PUBLICATION Copyright © 2012 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., Ill River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representation or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some Content that appears in print, however, may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Cataloging-in-Publication Data: Cox, David A. Galois theory / David A. Cox. 2nded. p. cm. Includes bibliographical references and index. ISBN 978-I-i 18-07205-9 (cloth) 1. Galois theory. I. Title. QA2i4.C69 2012 5l2'.32—dc23 2011039044 Printed inthe United States of America. 10 9 8 7 6 5 4 3 2 1 To my family, even the cats CONTENTS Preface to the First Edition xvii Preface to the Second Edition Xxi Notation xxiii Basic Notation xxiii 1 2 Chapter-by-Chapter Notation xxv PART POLYNOMIALS I 1 Cubic Equations 3 1.1 Cardan's Formulas 4 Historical Notes 8 1.2 Permutations of the Roots 10 A Permutations 10 B The Discriminant 11 C Symmetric Polynomials 13 VII Viii CONTENTS MathematicalNotes 14 Historical Notes 14 1.3 Cubic Equations over the Real Numbers 15 A The Number of Real Roots 15 B Trigonometric Solution of the Cubic 18 Historical Notes 19 References 23 2 Symmetric Polynomials 25 2.1 Polynomials of Several Variables 25 A The Polynomial Ring in n Variables 25 B The Elementary Symmetric Polynomials 27 Mathematical Notes 29 2.2 Symmetric Polynomials 30 A The Fundamental Theorem 30 B The Roots of a Polynomial 35 C Uniqueness 36 Mathematical Notes 37 Historical Notes 38 2.3 Computing with Symmetric Polynomials (Optional) 42 A Using Mat hematica 42 B Using Maple 44 2.4 The Discriminant 46 Mathematical Notes 48 Historical Notes 50 References 53 3 Roots of Polynomials 55 3.1 The Existence of Roots 55 Mathematical Notes 59 Historical Notes 61 3.2 The Fundamental Theorem of Algebra 62 Mathematical Notes 66 Historical Notes 67 References 70
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