Galois Theory
Classical Galois theory is a subject generally acknowledged to be one of the most central and beautiful areas in pure mathematics. This text develops the subject systematically and from the beginning, requiring of the reader only basic facts about polynom
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S. Axler K.A. Ribet
Steven H. Weintraub
Galois Theory
Springer
Steven H. Weintraub Department of Mathematics Lehigh University Bethlehem, PA 18015 USA [email protected]
Editorial Board (North America): S. Axler Mathematics Department San Francisco State University San Francisco, CA 94132 USA [email protected]
K.A. Ribet Mathematics Department University of California at Berkeley Berkeley, CA 94720-3840 USA [email protected]
Mathematics Subject Classification 2000: 12-01, 12fl0, 11R32 Library of Congress Control Number: 2005932841 ISBN-10: 0-387-28725-6 ISBN-13: 978-0387-28725-6
e-ISBN: 0-387-28917-8
Printed on acid-free paper.
©2006 Springer Science+Business Media, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media Inc., 233 Spring Street. New York. NY 10013, U.S.A.), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed in the United States of America. 987654321 springeronline.com
(TXQ/MP)
To Judy, after 14 years, and to Blake
Contents
Preface
ix
1
Introduction to Galois Theory 1.1 Some Introductory Examples
1 1
2
Field Theory and Galois Theory 2.1 Generalities on Fields 2.2 Polynomials 2.3 Extension Fields 2.4 Algebraic Elements and Algebraic Extensions 2.5 Splitting Fields 2.6 Extending Isomorphisms 2.7 Normal, Separable, and Galois Extensions 2.8 The Fundamental Theorem of Galois Theory 2.9 Examples 2.10 Exercises
7 7 11 15 18 22 24 25 29 37 39
3
Development and Applications of Galois Theory 3.1 Symmetric Functions and the Symmetric Group 3.2 Separable Extensions 3.3 Finite Fields 3.4 Disjoint Extensions 3.5 Simple Extensions 3.6 The Normal Basis Theorem 3.7 Abelian Extensions and Kummer Fields 3.8 The Norm and Trace 3.9 Exercises
45 45 51 54 57 63 66 70 76 79
viii
Contents
4
Extensions of the Field of Rational Numbers 4.1 Polynomials in Q[X] 4.2 Cyclotomic Fields 4.3 Solvable Extensions and Solvable Groups 4.4 Geometric Constructions 4.5 Quadratic Extensions of Q 4.6 Radical Polynomials and Related Topics 4.7 Galois Groups of Extensions of Q 4.8 The Discriminant 4.9 Practical Computation of Galois Groups 4.10 Exercises
85 85 89 93 97 103 108 118 124 127 133
5
Further Topics in Field Theory 5.1 Separable and Inseparable Extensions 5.2 Normal Extensions 5.3 The Algebraic Cslosure 5.4 Infinite Galois Extensions 5.5 Exercises
139 139 147 151 156 167
A
Some Results from Group Theory A.l Solvable Groups A.2 /^-Groups A.3 Symmetric and Alternating Groups
169 169 173 174
B
A Lemma on Constructing Field
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