Galois Theory, Second Edition

603 Pages·2012·4.99 MB·English

by  David A. Cox

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Monographs
Theory
Galois
Second Edition
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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
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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