Elements of Abstract Algebra

256 Pages·1984·4.37 MB·English

by  Allan Clark

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For my parents
I was just going to say, when I was interrupted, that one of the many ways of classifying minds is
under the heads of arithmetical and algebraical intellects. All economical and practical wisdom is an
extension of the following arithmetical formula: 2 + 2 = 4. Every philosophical proposition has the
more general character of the expression a + b = c. We are mere operatives, empirics, and egotists
until we learn to think in letters instead of figures.
OLIVER WENDELL HOLMES
The Autocrat of the Breakfast Table
Copyright © 1971, 1984 by Allan Clark.
All rights reserved.
This Dover edition, first published in 1984, is an unabridged and corrected republication of the work
first published by Wadsworth Publishing. Company, Belmont, California, in 1971.
Library of Congress Cataloging in Publication Data
Clark, Allan, 1935–
Elements of abstract algebra.
“Corrected republication”—Verso t.p.
Originally published: Belmont, Calif. : Wadsworth, © 1971.
Bibliography: p.
Includes index.
1. Algebra, Abstract. 1. Title.
[QA162.C57 1984] 512′.02 84-6118
9780486140353
Manufactured in the United States by Courier Corporation
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Foreword
Modern or “abstract” algebra is widely recognized as an essential element of mathematical
education. Moreover, it is generally agreed that the axiomatic method provides the most elegant and
efficient technique for its study. One must continually bear in mind, however, that the axiomatic
method is an organizing principle and not the substance of the subject. A survey of algebraic
structures is liable to promote the misconception that mathematics is the study of axiom systems of
arbitrary design. It seems to me far more interesting and profitable in an introductory study of modern
algebra to carry a few topics to a significant depth. Furthermore I believe that the selection of topics
should be firmly based on the historical development of the subject.
This book deals with only three areas of abstract algebra: group theory, Galois theory, and classical
ideal theory. In each case there is more depth and detail than is customary for a work of this type.
Groups were the first algebraic structure characterized axiomatically. Furthermore the theory of
groups is connected historically and mathematically to the Galois theory of equations, which is one of
the roots of modern algebra. Galois theory itself gives complete answers to classical problems of
geometric constructibility and solvability of equations in radicals. Classical ideal theory, which arose
from the problems of unique factorization posed by Fermat’s last theorem, is a natural sequel to
Galois theory and gives substance to the study of rings. All three topics converge in the fundamental
theorem of algebraic number theory for Galois extensions of the rational field, the final result of the
book.
Emil Artin wrote: We all believe that mathematics is an art. The author of a book, the lecturer in a
classroom tries to convey the structural beauty of mathematics to his readers, to his listeners. In this
attempt he must always fail. Mathematics is logical to be sure; each conclusion is drawn from
previously derived statements. Yet the whole of it, the real piece of art, is not linear; worse than that
its perception should be instantaneous. We all have experienced on some rare occasions the feeling of
elation in realizing that we have enabled our listeners to see at a moment’s glance the whole
architecture and all its ramifications. How can this be achieved? Clinging stubbornly to the logical
sequence inhibits visualization of the whole, and yet this logical structure must predominate or chaos
would result. 1
A text must cling stubbornly to the logical sequence of the subject. A lecturer may be peripatetic,
frequently with engaging results, but an author must tread a straight and narrow path. However,
though written sequentially, this book need not be read that way. The material is broken into short
articles, numbered consecutively throughout. These can be omitted, modified, postponed until needed,
or given for outside reading. Most articles have exercises, a very few of which are used later in proofs.
What can be covered in an ordinary course and for what students the text is suitable are questions left
to the instructor, who is the best judge of local conditions. It is helpful, but certainly not essential, for
the reader to know a little linear algebra for the later chapters—in particular Cramer’s rule. (Vector
spaces, bases, and dimension are presented in articles 90–95.)
Finally, I must gratefully acknowledge the assistance of Mrs. Theodore Weller and Miss Elizabeth
Reynolds, who typed the manuscript, and the help of Messrs. George Blundall and John Ewing, who
gave their time and patience to proofing it.
Providence, Rhode Island
January 1, 1970
Table of Contents
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Title Page
Dedication
Copyright Page
Foreword
Introduction
Chapter 1 - Set Theory
Chapter 2 - Group Theory
Chapter 3 - Field Theory
Chapter 4 - Galois Theory
Chapter 5 - Ring Theory
Chapter 6 - Classical Ideal Theory
Bibliography
Index