Table Of ContentA COURSE IN
Galois theory
D.J .H.G ARLING
Reader in Mathematical Analysis, University of Cambridge
and Fellow ofSt John's College, Cambridge
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CAMBRIDGE UNIVERSITY PRESS
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©Cambridge University Press 1986
First published 1986
Printed in Great Britain at the University Press, Cambridge
British Library cataloguing in publication data
Garling, D.J.H.
Galois theory.
1. Galois theory
I. Title
512'.32 QA211
Library of Congress cataloguing in publication data
Garling, D. J. H.
Galois theory.
1. Galois theory. I. Title.
QA17l.G277 1986 512'.32 86-6081
ISBN 0 521 32077 1 hard covers
ISBN 0 521 31249 3 paperback
MP
CONTENTS
1---------
MATH
GARLING,
QA A COURSE
1 71 IN GALOIS
G277
*********
1986 MATH.
LIBRARY
Preface vii
Part 1: Algebraic preliminaries 1
1 Groups, fields and vector spaces 3
1.1. Groups 3
1.2. Fields 8
1.3. Vector spaces 9
2 The axiom of choice, and Zorn's lemma 14
2.1. The axiom of choice 14
2.2. Zorn's lemma 14
2.3. The existence of a basis 15
3 Rings 18
3.1. Rings 18
3.2. Integral domains 20
3.3. Ideals 21
3.4. Irreducibles, primes and unique factorization domains 24
3.5. Principal ideal domains 27
3.6. Highest common factors 29
3.7. Polynomials over unique factorization domains 31
3.8. The existence ofmaximal proper ideals 34
3.9. More about fields 35
Part 2: The theory of fields, and Galois theory 37
4 Field extensions 39
4.1. Introduction 39
4.2. Field extensions 40
4.3. Algebraic and transcendental elements 42
4.4. Algebraic extensions 46
4.5. M onomorphisms of algebraic extensions 48
5 Tests for irreducibility 49
5.1. Introduction 49
iv Contents
5.2. Eisenstein's criterion 51
5.3. Other methods for establishing irreducibility 52
6 Ruler-and-compass constructions 54
6.1. Constructible points 54
6.2. The angle n/3 cannot be trisected 57
6.3. Concluding remarks 58
7 Splitting fields 59
7.1. Splitting fields 60
7.2. The extension of monomorphisms 62
7.3. Some examples 67
8 The algebraic closure of a field 71
8.1. Introduction 71
8.2. The existence of an algebraic closure 72
8.3. The uniqueness of an algebraic closure 75
8.4. Conclusions 77
9 Normal extensions 78
9.1. Basic properties 78
9.2. Monomorphisms and automorphisms 80
10 Separability 82
10.1. Basic ideas 82
10.2. Monomorphisms and automorphisms 83
10.3. Galois extensions 84
10.4. Differentiation 85
10.5. The Frobenius monomorphism 87
10.6. Inseparable polynomials 88
11 Automorphisms and fixed fields 91
11.1. Fixed fields and Galois groups 91
11.2. The Galois group of a polynomial 94
11.3. An example 96
11.4. The fundamental theorem of Galois theory 97
11.5. The theorem on natural irrationalities 99
12 Finite fields 101
12 .1. A description of the finite fields 101
12.2. An example 102
12.3. Some abelian group theory 103
12.4. The multiplicative group of a finite field 105
12.5. The automorphism group of a finite field 105
13 The theorem of the primitive element 107
13.1. A criterion in terms of intermediate fields 107
13.2. The theorem of the primitive element 108
13.3. An example 109
14 Cubics and quartics 110
14.1. Extension by radicals 110
14.2. The discriminant 111
14.3. Cubic polynomials 113
Contents v
14.4. Quartic polynomials 115
15 Roots of unity 118
15 .1. Cyclotomic polynomials 118
15.2. Irreducibility 120
15.3. The Galois group of a cyclotomic polynomial 121
16 Cyclic extensions 123
16.1. A necessary condition 123
16.2. Abel's theorem 124
16.3. A sufficient condition 125
16.4. Kummer extensions 128
17 Solution by radicals 131
17.1. Soluble groups: examples 131
17.2. Soluble groups: basic theory 132
17.3. Polynomials with soluble Galois groups 134
17.4. Polynomials which are solvable by radicals 135
18 Transcendental elements and algebraic independence 139
18.1. Transcendental elements and algebraic independence 139
18.2. Transcendence bases 141
18.3. Transcendence degree 143
18.4. The tower law for transcendence degree 144
18.5. Luroth's theorem 145
19 Some further topics 147
19.1. Generic polynomials 147
19.2. The normal basis theorem 150
19.3. Constructing regular polygons 152
20 The calculation of Galois groups 155
20.1. A procedure for determining the Galois group of a polynomial 155
20.2. The soluble transitive subgroups of EP · 158
20.3. The Galois group of a quintic 161
20.4. Concluding remarks 162
Index 163
PREFACE
Galois theory is one of the most fascinating and enjoyable branches of
algebra. The problems with which it is concerned have a long and
distinguished history: the problems of duplicating a cube or trisecting an
angle go back to the Greeks, and the problem of solving a cubic, quartic or
quintic equation to the Renaissance. Many of the problems that are raised
are of a concrete kind (and this, surely, is why it is so enjoyable) and yet the
needs of the subject have led to substantial development in many branches
of abstract algebra: in particular, in the theory of fields, the theory of groups,
the theory of vector spaces and the theory of commutative rings.
In this book, Galois theory is treated as it should be, as a subject in its
own right. Nevertheless, in the process, I have tried to show its relationship
to various topics in abstract algebra: an understanding of the structures of
abstract algebra helps give a shape to Galois theory and conversely Galois
theory provides plenty of concrete examples which show the point of
abstract theory.
This book comprises two unequal parts. In the first part, details are given
of the algebraic background knowledge that it is desirable to have before
beginning to study Galois theory. The first chapter is quite condensed: it is
intended to jog the memory, to introduce the terminology and notation
that is used, and to give one or two examples which will be useful later. In
the second chapter, the axiom of choice and Zorn's lemma are described.
Algebra is principally concerned with finite discrete operations, and it
would have been possible, at the cost of not establishing the existence of
algebraic closures, to have avoided all use of the axiom of choice.
Mathematicians do, however, need to know about the axiom of choice, and
this is an appropriate place to introduce it. A reader who has not met the
concepts of this chapter before may omit it (and Chapter 8); preferably, he
or she should read through it quite quickly to get some idea of the issues
viii Preface
involved, and not worry too much about the details. The third chapter, on
rings, is much more important, and should be read rather carefully. It is an
important fact that polynomials with integer coefficients and polynomials
in several variables enjoy unique factorization, and it is necessary to go
beyond principal ideal domains to establish this fact. There are some other
special results from abstract algebra that are needed (such as basic
properties of soluble groups): most of these are established when the need
arises.
The second, more substantial, part is concerned with the theory of fields
and with Galois theory,and contains the main material of the book. Of its
nature, the theory develops an inexorable momentum. Nevertheless, there
are many digressions (for example, concerning irreducibility, geometric
constructions, finite fields and the solution of cubic and quartic equations):
one of the pleasures of Galois theory is that there are many examples which
illustrate and depend upon the general theory, but which also have an
interest of their own. The high point of the book is of course the resolution
of the problem of when a polynomial is soluble by radicals. I have, however,
tried to emphasize (in the final chapter in particular) that this is not the end
of the story: the resolution of the problem raises many new problems, and
Galois theory is still a lively subject.
Two hundred exercises are scattered through the text. It has been
suggested to me that this is rather few: I think that anyone who honestly
tries them all will disagree! In my opinion, text-book exercises are often too
straightforward, but some of these exercises are quite hard. The successful
solution of a challenging problem gives a much better understanding of the
powers and limitations of the theory than any number of trivial ones.
Remember that mathematics is not a spectator sport!
This book grew out of a course of lectures which I gave for several years
at Cambridge University. I have, however, not resisted the temptation to
add extra material. A shorter course than the whole book provides can be
obtained by omitting Chapter 2, Chapter 8, Section 10.6, Section 18.5 and
Chapter 20. I am grateful to all who attended the course, and helped me to
improve it. I am particularly grateful to Robert J. H. A. Turnbull, who read
and commented helpfully on an early draft and also detected many errors in
the final version of this book.
PARTJ
Algebraic preliminaries
