A Classical Introduction to Galois Theory

297 Pages·2012·4.043 MB·English

by  Stephen C. Newman

A CLASSICAL
INTRODUCTION
TO GALOIS THEORY
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A CLASSICAL
INTRODUCTION
TO GALOIS THEORY
STEPHEN C. NEWMAN
UniversityofAlberta,
Edmonton,Alberta,Canada
AJOHNWILEY&SONS,INC.,PUBLICATION
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LibraryofCongressCataloging-in-PublicationData:
Newman,StephenC.,1952–
AclassicalintroductiontoGaloistheory/StephenC.Newman.
p.cm.
Includesindex.
ISBN978-1-118-09139-5(hardback)
1. Galoistheory. I. Title.
QA214.N492012
512(cid:2).32–dc23
2011053469
PrintedintheUnitedStatesofAmerica
10987654321
ToSandra
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CONTENTS
PREFACE xi
1 CLASSICALFORMULAS 1
1.1 Quadratic Polynomials / 3
1.2 Cubic Polynomials / 5
1.3 Quartic Polynomials / 11
2 POLYNOMIALSANDFIELDTHEORY 15
2.1 Divisibility / 16
2.2 Algebraic Extensions / 24
2.3 Degree of Extensions / 25
2.4 Derivatives / 29
2.5 Primitive Element Theorem / 30
2.6 Isomorphism Extension Theorem and Splitting Fields / 35
3 FUNDAMENTALTHEOREMONSYMMETRIC
POLYNOMIALSANDDISCRIMINANTS 41
3.1 Fundamental Theorem on Symmetric Polynomials / 41
3.2 Fundamental Theorem on Symmetric Rational Functions / 48
3.3 Some Identities Based on Elementary Symmetric
Polynomials / 50
vii
viii CONTENTS
3.4 Discriminants / 53
3.5 Discriminants and Subfields of the Real Numbers / 60
4 IRREDUCIBILITYANDFACTORIZATION 65
4.1 Irreducibility Over the Rational Numbers / 65
4.2 Irreducibility and Splitting Fields / 69
4.3 Factorization and Adjunction / 72
5 ROOTSOFUNITYANDCYCLOTOMICPOLYNOMIALS 80
5.1 Roots of Unity / 80
5.2 Cyclotomic Polynomials / 82
6 RADICALEXTENSIONSANDSOLVABILITYBY
RADICALS 89
6.1 Basic Results on Radical Extensions / 89
6.2 Gauss’s Theorem on Cyclotomic Polynomials / 93
6.3 Abel’s Theorem on Radical Extensions / 104
6.4 Polynomials of Prime Degree / 109
7 GENERALPOLYNOMIALSANDTHEBEGINNINGSOF
GALOISTHEORY 117
7.1 General Polynomials / 117
7.2 The Beginnings of Galois Theory / 124
8 CLASSICALGALOISTHEORYACCORDINGTOGALOIS 135
9 MODERNGALOISTHEORY 151
9.1 Galois Theory and Finite Extensions / 152
9.2 Galois Theory and Splitting Fields / 156
10 CYCLICEXTENSIONSANDCYCLOTOMICFIELDS 171
10.1 Cyclic Extensions / 171
10.2 Cyclotomic Fields / 179
11 GALOIS’SCRITERIONFORSOLVABILITY
OFPOLYNOMIALSBYRADICALS 185
12 POLYNOMIALSOFPRIMEDEGREE 192
CONTENTS ix
13 PERIODSOFROOTSOFUNITY 200
14 DENESTINGRADICALS 225
15 CLASSICALFORMULASREVISITED 231
15.1 General Quadratic Polynomial / 231
15.2 General Cubic Polynomial / 233
15.3 General Quartic Polynomial / 236
APPENDIXA COSETSANDGROUPACTIONS 245
APPENDIXB CYCLICGROUPS 249
APPENDIXC SOLVABLEGROUPS 254
APPENDIXD PERMUTATIONGROUPS 261
APPENDIXE FINITEFIELDSANDNUMBERTHEORY 270
APPENDIXF FURTHERREADING 274
REFERENCES 277
INDEX 281