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exploratory galois theory
Combiningaconcreteperspectivewithanexploration-basedapproach,
ExploratoryGaloisTheorydevelopsGaloistheoryatanentirelyundergraduate
level.Thetextgroundsthepresentationintheconceptofalgebraicnumbers
withcomplexapproximationsandassumesofitsreadersonlyafirstcoursein
abstractalgebra.Theauthororganizesthetheoryaroundnaturalquestions
aboutalgebraicnumbers,andexerciseswithhintsandproofsketches
encouragestudents’participationinthedevelopment.Forreaderswith
MapleorMathematica,thetextintroducestoolsforhands-onexperimentation
withfiniteextensionsoftherationalnumbers,enablingafamiliaritynever
beforeavailabletostudentsofthesubject.ExploratoryGaloisTheoryincludes
classicalapplications,fromruler-and-compassconstructionstosolvabilityby
radicals,andalsooutlinesthegeneralizationfromsubfieldsofthecomplex
numberstoarbitraryfields.Thetextisappropriatefortraditionallecture
courses,forseminars,orforself-pacedindependentstudybyundergraduates
andgraduatestudents.
JohnSwallowisJ.T.KimbroughAssociateProfessorofMathematicsat
DavidsonCollege.HeholdsadoctoratefromYaleUniversityforhisworkin
Galoistheory.Heistheauthororco-authorofadozenarticles,includingan
essayinTheAmericanScholar.HisworkhasbeensupportedbytheNational
ScienceFoundation,theNationalSecurityAgency,andtheAssociatedColleges
oftheSouth.
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Exploratory
Galois Theory
JOHN SWALLOW
DavidsonCollege
iii
cambridge university press
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
cb2 2ru
The Edinburgh Building, Cambridge , UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521836500
© John Swallow 2004
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2004
isbn-13 978-0-511-22932-9
eBook (ebrary)
isbn-10 0-511-22932-1
eBook (ebrary)
isbn-13 978-0-521-83650-0
hardback
isbn-10 0-521-83650-6
hardback
isbn-13 978-0-521-54499-3 paperback
isbn-10 0-521-54499-8
paperback
url
Cambridge University Press has no responsibility for the persistence or accuracy of s
for external or third-party internet websites referred to in this publication, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
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Contents
Preface pageix
Introduction 1
1 Preliminaries 5
§1. Polynomials,PolynomialRings,Factorization,andRootsinC 5
§2. ComputationwithRootsandFactorizations:MapleandMathematica 12
§3. RingHomomorphisms,Fields,Monomorphisms,andAutomorphisms 15
§4. Groups,Permutations,andPermutationActions 17
§5. Exercises 18
2 AlgebraicNumbers,FieldExtensions,andMinimalPolynomials 22
§6. ThePropertyofBeingAlgebraic 22
§7. MinimalPolynomials 23
§8. TheFieldGeneratedbyanAlgebraicNumber 25
§9. ReducedFormsinQ(α):MapleandMathematica 33
§10. Exercises 35
3 WorkingwithAlgebraicNumbers,FieldExtensions,andMinimalPolynomials 39
§11. MinimalPolynomialsAreAssociatedtoWhichAlgebraicNumbers? 39
§12. WhichAlgebraicNumbersGenerateaGeneratedField? 42
§13. ExerciseSet1 49
§14. ComputationinAlgebraicNumberFields:MapleandMathematica 51
§15. ExerciseSet2 61
4 MultiplyGeneratedFields 63
§16. FieldsGeneratedbySeveralAlgebraicNumbers 63
§17. CharacterizingIsomorphismsbetweenFields:ThreeCubicExamples 72
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viii Contents
§18. IsomorphismsfromMultiplyGeneratedFields 78
§19. FieldsandSplittingFieldsGeneratedbyArbitrarilyManyAlgebraic
Numbers 83
§20. ExerciseSet1 86
§21. ComputationinMultiplyGeneratedFields:MapleandMathematica 89
§22. ExerciseSet2 100
5 TheGaloisCorrespondence 103
§23. NormalFieldExtensionsandSplittingFields 103
§24. TheGaloisGroup 105
§25. InvariantPolynomials,GaloisResolvents,andtheDiscriminant 115
§26. ExerciseSet1 127
§27. DistinguishingNumbers,DeterminingGroups 128
§28. ComputationofGaloisGroupsandResolvents:Maple andMathematica 137
§29. ExerciseSet2 149
6 SomeClassicalTopics 152
§30. RootsofUnityandCyclotomicExtensions 152
§31. CyclicExtensionsoverFieldswithRootsofUnity 156
§32. BinomialEquations 161
§33. Ruler-and-CompassConstructions 163
§34. SolvabilitybyRadicals 171
§35. Characteristicp andArbitraryFields 177
§36. FiniteFields 186
HistoricalNote 193
Appendix:SubgroupsofSymmetricGroups 197
§1. TheSubgroupsofS 197
4
§2. TheSubgroupsofS 198
5
Bibliography 201
Index 205
