Rings, Fields and Groups, An Introduction to Abstract Algebra

410 Pages·1991·13.9 MB·English

by  Reg Allenby

Rings, Fields and
Groups
An Introduction to Abstract Algebra
R B J T Allenby
Senior Lecturer, School of Mathematics, University of Leeds
Edward Arnold
A division of I Iodder & Stotugliton
I10N1)ON NEW YORK ME AI'CKLANI)
For Janet,
Elizabeth and
Rachel
© 1991 R. B. J. T. Allenby
First published in Great Britain 1983
Reprinted with corrections 1985 and 1986
Reprinted 1988. 1989
Second edition 1991
Distributed in the USA by Routledge, Chapman and Hall. inc..
29 West 35th Street. New York, NY 10001
Britich Library Cataloguing Publication Data
Allenby. R. B. J. T.
Rings, fields and groups.
1. Rings (Algebra)
1. Title
512'.4 QC251
ISBN 0-7131-3476-3
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Printed in Great Britain for Edward Arnold,
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by St Edmundsbury Press Ltd. Bury St Edmunds, Suffolk
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Preface to the first edition
overall aim of this book is to present a fairly leisurely introduction to
The
some of the results, methods and ideas which are increasingly to be found in
first and second year abstract algebra courses in British universities and
polytechnics and in equivalent courses elsewhere.
There are several ways in which an author might present such an introduc-
tion. One is the (to some) aesthetically most pleasing take-it-or-leave-it purely
axiomatic approach in which the reader is given a list of the appropriate
definitions and is then led through proofs of those theorems universally agreed
by those already in the know to be basic to the subject. The guiding spirit
behind such an approach would almost certainly force the author to begin
with a long and rather dry axiomatic development of the set-theoretical
language needed. Although the purity of this particular approach would
probably be preferred by a majority of practising pure mathematicians who
already have a degree of familiarity with the material, the present author has
found that it is appreciated by only a handful in every class of fifty or so
beginners. Many students become restless at such an approach as they find
difficulty in connecting the discussion with ideas with which they are already
familiar. The sudden change from the world of concrete examples usually
found in school mathematics to the abstract setting seemingly remote from the
real world is one which can lead some students away from abstract algebra in
particular and pure mathematics in general, a state of affairs which naturally
saddens the author who much enjoys sharing with his beginning students the
pleasure (even excitement!) to be obtained from following through some of the
clever ideas and neat arguments to be found there.
A second approach, possibly more attractive to the beginner, would be to
present a detailed historical account of the develolpment of algebra from, say,
1500 to the present day. To the student who is aware of some of the upheavals
in school mathematics courses in recent years, the inclusion of an account of
some of those theories which were once vigorously developed and were
expected to become important but have generally failed to find favour or
application might prove especially interesting! However, such an approach
would leave, in a volume of reasonable size, little room for a really detailed
account of any of the theories reviewed.
An intermediatecourse, and the one taken here, is to try to get fairly quickly
into the spirit of abstract algebra, whilst at the same time interjecting
occasional remarks and comments, either because oftheir historical content
iv Preface
or because, quite simply, the author thinks they are fascinating (or both!). In
particular, several complete sections are included more as light reading than as
essential material. Amongst these I include Sections 3.5, 3.9, 4.4, 5.2, 5.12 and
6.7. The present approach, therefore, is a mixture of the formal and the
informal. The author has certainly found this mixture acceptable in courses he
has given on both sides of the Atlantic. On the one hand we take an informal
approach to set theory because our concern is with the algebra; on the other
hand we do not want to throw all formal working to the wind, present day
algebra being, as it is, an axiomatic discipline. Indeed, one of the chief aims of
the book is to develop the reader's critical faculties and we believe it preferable
to begin this in Chapter 1 where, because of the student's (intuitive) familiarity
with much of its content, the critical approach seems all the more prominent.
(See also the Problems posed at the end of the Prologue.) Here formal
definitions are given when some readers might feel that informal ones would
suffice. (See, for instance, the development of the idea of polynomial from
Sections 1.6 to 1.8). Such readers should not need encouraging to read,
carefully, the reasons put forward for preferring the more formal approach.
We also use this chapter to try and answer, by example, a question often asked
by beginners, namely 'How much proof should I give?' The answer clearly
depends upon the knowledge and maturity of the people trying to correspond.
To help the reader through his first encounter with several of the proofs the
author has been more expansive than he would normally be in communicating
with a colleague. Those extra portions, which can be omitted as confidence
grows and which to some extent are the answers to the questions the reader
should be asking himself as he works through the text, have been put in square
brackets.
We begin with a prologue in which we attempt to answer some of the
questions students seem afraid to ask: What is abstract algebra? How did it
develop? What use is it? The historical account of the development of algebra
will include many words not familiar to the beginner, but we feel that in a new
land it is preferable to possess a map, even one in a foreign language, than no
map at all. In placing this material before rather than after the main body of
the text we hope to whet the reader's appetite and heighten his sense of
excitement with a description of the discoveries and inventions made by some
of the mathematical giants of the past and that this excitement will fire him
sufficiently to read this book avidly even when (as it probably will) the going
gets a bit difficult.
The numbering of Chapter 0 indicates that we view it as a preliminary to the
text proper. Chapters 1, 2, 3 and 4 concentrate on algebraic systems known as
rings and fields (though these names are not formally introduced until Chapter
3) the concept of group not being mentioned until Chapter 5. A majority of
texts on abstract algebra offer a study of the theory of groups before a study of
rings, the reason often given being that groups, having only one binary
operation, are simpler to begin with than are rings and fields which have two.
The author (a group theorist!) feels that there is a rather strong case for
Preface v
this order; the fact is that natural concrete examples of rings and
reversing
fields (the integers, polynomials, the rational, real and complex numbers) are
much better known to the beginner than are the equivalent concrete examples
of groups (mainly symmetries of 2- and 3-dimensional figures). (This author
will just not accept the complaint 'But the integers form a group under
addition': so they do but that is not the natural way to look at them. Indeed
the author rebels strongly against the argument which, briefly, runs: 'The
integers under addition form a group. Therefore we must study group
theory.') What in the author's opinion really clinches the argument for
studying rings and fields before groups are the several exciting applications
that can quite quickly be made to easily stated yet non-trivial problems in the
theory of numbers and of geometrical constructions. (See especially Sections
3.8 and 4.6.)
The placing of ring theory before group theory will, it is true, give rise to a
little more repetition of corresponding elementary concepts than might have
been the case with the more usual presentation. The author does not, however,
feel any need to apologise for that! (On the other hand Chapters 5and 6 make
little essential use of Chapters 3 and 4, so they can be studied directly after
Chapter 2.)
Throughout the text problems, numbered only for ease of reference, have
been inserted as they have occurred to the author. Some are reasonably easy,
some solved later in the text and some are quite hard. I leave you to find out
which! The purpose of these problems is (i) to set you thinking and then
discussing them with a colleague or teacher; (ii) to get you into the habit of
posing questions of this kind to yourself. Active participation is always much
more exciting (and instructional) than passive reading!
In this book the numbering of theorems, lemmas, etc. in any one chapter is
consecutive, thus: Theorem 5.5.4, Example 5.5.5, Notation 5.5.6, Definition
5.5.7. When referring to a theorem, lemma, etc. given elsewhere in the text
usually only its number is given (e.g. 3.8.2). Reference to an exercise is
however given in full (e.g. exercise 3.2.14) except when the exercise referred to
is at the end of the section concerned. Thus, within Section 3.2, exercise 3.2.14
would be referred to as 'exercise 14'.
In producing this text I have received help from several people, especially
from the secretaries in the School of Mathematics in the University of Leeds.
In particular I should like to thank Mrs M M Turner, Mrs P Jowett, Mrs A
Landford and Mrs M R Williams. Several colleagues in Leeds and elsewhere
have offered helpful comments and gentle criticism on parts of the manuscript.
Here I especially with to thank Drs J C McConnell, E W Wallace and J R
Ravetz. For supplying me with photographs I thank the keeper of the David
Eugene Smith collection at Columbia University, New York, and especially
Prof Dr Konrad Jacobs of the University of Erlangen—Nürnberg who kindly
donated the pictures of Emmy Noether and Richard Dedekind.
Leeds RBJTA
1982

Contents
Preface to the first edition
Preface to the second edition vi
How to read this book
Prologue
0 Elementary set theory and methods of proof
1
0.1 Introduction
0.2 Sets
0.3 Newsets from old 3
0.4 Some methods of proof
5
I Numbers and polynomials 10
1.1 Introduction 10
1.2 The basic axioms. Mathematical induction 10
1.3 Divisibility, irreducibles and primes in 1 20
Biography and portrait of Hubert 25
1.4 GCDs 26
1.5 The unique factorisation theorem (two proofs) 33
1.6 Polynomials—what are they? 35
1.7 The basic axioms 37
1.8 The 'new' notation 39
1.9 Divisibility, irreducibles and primes in Q[x] 42
1.10 The division algorithm 49
1.11 Roots and the remainder theorem
51
2 Binary relations and binary operations 57
2.1 Introduction 57
2.2 Congruence mod n. Binary relations 57
2.3 Equivalence relations and partitions 61
2.4 63
Biography and portrait of Gauss 68
2.5 Some deeper number-theoretic results concerning
congruences 69
viii Contents
2.6 Functions
2.7 Binary operations
3 Introduction to rings
3.1 Introduction
3.2 The abstract definition of a ring
Biography and portrait of Hamilton
3.3 Ring properties deducible from the axioms
3.4 Subrings, subfields and ideals
Biography and portrait of Noether
Biography and portrait of Fermat
3.5 Fermat's conjecture (FC)
3.6 Divisibility in rings
3.7 Euclidean rings, unique factorisation domains and principal
ideal domains
3.8 Three number-theoretic applications
Biography and portrait of Dedekind
3.9 Unique factorisation reestablished. Prime and maximal ideals
3.10 Isomorphism. Fields of fractions. Prime subfields
3.11 U[x] where U is a UFD
3.12 Ordered domains. The uniqueness of Z
4 Factor rings and fields
4.1 Introduction
4.2 Return to roots. Ring homomorphisms. Kronecker's theorem
4.3 The isomorphism theorems
4.4 Constructions of R from Q and of C from R
Biography and portrait of Cauchy
4.5 Finite fields
Biography and portrait of Moore
4.6 Constructions with compass and straightedge
4.7 Symmetric polynomials
4.8 The fundamental theorem of algebra
5 Basic group theory
5.1 Introduction
5.2 Beginnings
Biography and portrait of Lagrange
5.3 Axioms and examples
5.4 Deductions from the axioms
5.5 The symmetric and the alternating groups
5.6 Subgroups. order of an element
5.7 Cosets of subgroups. Lagrange's theorem
Contents ix
5.8 Cyclic groups 213
5.9 Isomorphism. Group tables 216
Biography and portrait of Cayley 220
5.10 Homomorphisms. Normal subgroups 223
5.11 Factor groups. The first isomorphism theorem 229
5.12 Space groups and plane symmetry groups 233
6 Structure theorems of group theory 242
6.1 Introduction 242
6.2 Normaliser. Centraliser. Sylow's theorems 242
6.3 Direct products 250
6.4 Finite abelian groups 254
6.5 Soluble groups. Composition series 258
6.6 Some simple groups 268
7 A brief excursion into Galois theory 274
7.1 Introduction 274
Biography and portrait of Galois 275
7.2 Radical Towers and Splitting Fields 276
7.3 Examples 280
7.4 Some Galois groups: their orders and fixed fields 283
7.5 Separability and Normality 287
7.6 Subfields and subgroups 291
7.7 The groups Gal(R/F) and Gal(S1/F) 297
7.8 The groups - 300
7.9 A Necessary condition for the solubility of a polynomia
equation by radicals 302
Biography and portrait of Abel 303
7.10 A Sufficient condition for the solubility of a polynomial
equation by radicals 305
7.11 Non-soluble polynomials: grow your own! 308
7.12 Galois Theory—old and new 311
Partial solutions to the exercises 315
Bibliography 364
Notation 371
Index 373