Abstract Algebra and Solution by Radicals

225 Pages·1992·87.278 MB·English

by  John E. Maxfield, Margaret W. Maxfield

ABSTRACT ALGEBRA
AND SOLUTION
BY RADICALS
John E. Maxfield
M arsaret W. Maxfield
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ABSTRACT ALGEBRA
AND SOLUTION
BY RADICALS
John £ Maxfield
Dean of the Graduate School and University Research
Louisiana Tech University
Margaret W. Maxfield
Professor of Mathematics and Statistics
Louisiana Tech University
Dover Publications, Inc., New York
Copyright © 1971 by John E. Maxfield and Margaret W. Maxfield.
All rights reserved under Pan American and International Copyright
Conventions.
Published in Canada by General Publishing Company, Ltd., 30 Lesmill
Road, Don Mills, Toronto, Ontario.
Published in the United Kingdom by Constable and Company, Ltd., 3 The
Lanchesters, 162-164 Fulham Palace Road, London W6 9ER.
This Dover edition, first published in 1992, is an unabridged, corrected
republication of the second (corrected) printing of the work first published by
the W. B. Saunders Company, Philadelphia, 1971.
Manufactured in the United States of America
Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501
Library of Congress Cataloging-in-Publication Data
Maxfield, John E. (John Edward), 1927-
Abstract algebra and solution by radicals / John E. Maxfield and Margaret
W. Maxfield.
p. cm.
“This Dover edition, first published in 1992, is an unabridged, corrected
republication of the second (corrected) printing of the work first published by
the W. B. Saunders Company, Philadelphia, 1971“—T.p. verso.
Includes index.
ISBN 0-486-67121-6
1. Algebra, Abstract. I. Maxfield, Margaret W., 1926- II. Title.
QA162.M39 1992
512'. 02—dc20 91-46292
CIP
DEDICATION
This book is lovingly and respectfully dedicated to our parents, C. G. and
Lucile Maxfield and Frederick V. and Irma W. Waugh, and to our other fine
teachers, including a neurotic magazine-straightener named Thompson, I
think, who kept drumming it into his seventh grade science class that precise
statement was the key to science, to Forrest H. Keck, to Mr. Dresia, and to the
memory of Mr. Herner, to the memory of Professor C. H. Yeaton, who gave
the most intriguing mathematics tests ever devised, to the memory of Professor
C. C. MacDuffee, who handed the chalk to his students, and to Professor Ivan
Niven, who suffered not one but two dissertations at our hands.
In the intensive editorial development carried out on the book we received
valuable help from Professors Arlo W. Schurle of Indiana University, Allen
B. Altman of the University of California at San Diego, Herbert J. Nichol
of Drexel University, Guy T. Hogan of SUNY State College at Oneonta, and
Brother Hugh N. Albright of LaSalle College, each of whom critically reviewed
the entire manuscript. Their detailed and percipient comments resulted in many
improvements to the final manuscript. We are grateful to George Fleming,
our editor at W. B. Saunders Company, who encouraged and organized the
project from the first tentative ideas.

CONTENTS
HOW TO USE THIS BOOK......................................................... VÜ
THE NEGATIVE OBJECT OF THIS BOOK................................ ¡x
CHAPTER I
GROUPS ............................................................... I
Discussion of the Group Properties................................................ 3
Further Properties of Groups............................................................. 5
Examples of Groups........................................................................... 9
CHAPTER 2
OTHER ABSTRACT ALGEBRAS................................................... 18
Examples of Various Abstract Algebras........................................... 20
Countable Infinity............................................................................... 25
Equivalence and Other Relations.................................................... 26
Isomorphism ......................................................................................... 33
Comparisons Among Abstract Algebras........................................... 43
CHAPTER 3
MORE ABOUT GROUPS............................................................... 47
Permutations ......................................................................................... 49
Subgroups ............................................................................................. 59
Congruence and Cosets...................................................................... 62
Quotient Groups.................................................................................. 70
CHAPTER 4
MAPPINGS THAT PRESERVE RELATIONS
78
vi CONTENTS
CHAPTER 5
GROUPS OF PRIME ORDER; TWO ALTERNATING GROUPS 88
HISTORICAL INTERMISSION....................................................... 96
Some Comments on the Biographies.................................................. 100
CHAPTER 6
POLYNOMIALS.............................................................................. 104
Factoring Polynomials in I[x]............................................................. 116
CHAPTER 7
ALGEBRAIC FIELD EXTENSIONS............................................... 125
Algebraic Numbers and Transcendental Numbers.................. 126
CHAPTER 8
GALOIS THEORY............................................................................ 142
CHAPTER 9
RADICALS AND ROOTS OF UNITY.......................................... ISO
CHAPTER 10
SOLUTION BY RADICALS........................................................... 155
APPENDICES ................................................................................. 161
Appendix A—Q Has K0 Members................................................ 163
Appendix B—R Has 2No Members............................................. 164
Appendix C—CHas2Xo Members................................................ 166
Appendix D—Graphing Complex Numbers.................................. 167
Appendix E—A Lattice of Abstract Algebras............................. 173
Appendix F—Modular Arithmetic.................................................. 174
Appendix G—Graphical Group Representations . . ..................... 175
Appendix H—History of Mathematics........................................... 176
Appendix I—More About Field Extensions.................................. 177
Appendix J—More About Galois Theory.................................... 184
Appendix K—The Cyclic Group of nth Roots of Unity................ 189
Appendix L—Proof That a Solvable Equation Has a Solvable
Galois Group............................................................. 190
Appendix M—Impossible Constructions......................................... 194
ANSWERS TO SELECTED EXERCISES .................................. 199
INDEX............................................................................................ 205
HOW TO USE
THIS BOOK
The main text of this book is unusually full, especially for a mathematics
book, which is often so hard to read that an ordinary literate adult is reduced to
having it read to him by the instructor. We hope to encourage you to read for
yourself.
To keep you from missing the basic organization in the admittedly wordy
text, we have set off the definitions and theorems with brackets . These form
the skeleton of the book. When reviewing, simply go over this skeleton, re­
reading the explanations in the text where necessary.
You will notice that some sections of the text are shaded, as,
for example, some of the material on isomorphisms surrounding
Definition 2—15. These are the sections we consider part of your
general education in this Age of Automation. The whole book can
bring you to an appreciation of abstraction and of arbitrary
postulational systems, ideas which are central in automation. The
ideas explained in the shaded sections are especially important and
characteristic of the kind of mathematics we are covering. Most of
them lend themselves to ordinary discussion among laymen without
any lengthy introduction or special jargon. Their flavor will stay with
you even if you forget specific theorems. That flavor is especially
useful to a teacher, for although he may not teach such advanced
material, he will want to present elementary material using the most
modern approach in a way consistent with what the student will
learn if he goes farther.
vii
vin HOW TO USE THIS BOOK
Some exercises are labeled “Classroom Exercise.” Attentive participation
while the whole class works these exercises together will form a bridge between
passive listening and active homework.
The “Historical Intermission” can be used in several different ways. In
our own pilot classes we have encouraged students to’relax with it on their own,
discussing it if they seem to want to.
The Appendices can be used to further your interest along several different
lines barely suggested in the text. They offer an opportunity for outside work
to bolster a sagging grade. Some contain proofs postponed from various
chapters so as not to impede the continuity of thought on a first reading.