Abstract Algebra: Applications to Galois Theory, Algebraic Geometry and Cryptography

379 Pages·2011·1.57 MB·English

by  Gerhard Rosenberger

De Gruyter Graduate

Celine Carstensen
Benjamin Fine
Gerhard Rosenberger
Abstract Algebra
Applications to Galois Theory,
Algebraic Geometry and Cryptography
De Gruyter
MathematicsSubject Classification2010: Primary:12-01,13-01, 16-01,20-01; Secondary:01-01,
08-01,11-01,14-01,94-01.
ThisbookisVolume11oftheSigmaSeriesinPureMathematics,HeldermannVerlag.
ISBN 978-3-11-025008-4
e-ISBN 978-3-11-025009-1
LibraryofCongressCataloging-in-PublicationData
Carstensen,Celine.
Abstract algebra : applications to Galois theory, algebraic geo-
metry, and cryptography / by Celine Carstensen, Benjamin Fine,
andGerhardRosenberger.
p.cm.(cid:2)(Sigmaseriesinpuremathematics;11)
Includesbibliographicalreferencesandindex.
ISBN978-3-11-025008-4(alk.paper)
1.Algebra,Abstract. 2.Galoistheory. 3.Geometry,Algebraic.
4.Crytography. I.Fine,Benjamin,1948(cid:2) II.Rosenberger,Ger-
hard. III.Title.
QA162.C375 2011
5151.02(cid:2)dc22
2010038153
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableintheInternetathttp://dnb.d-nb.de.
”2011WalterdeGruyterGmbH&Co.KG,Berlin/NewYork
Typesetting:Da-TeXGerdBlumenstein,Leipzig,www.da-tex.de
Printingandbinding:AZDruckundDatentechnikGmbH,Kempten
(cid:3)Printedonacid-freepaper
PrintedinGermany
www.degruyter.com
Preface
Traditionally,mathematicshasbeenseparatedintothreemainareas;algebra,analysis
and geometry. Of course there is a great deal of overlap between these areas. For
example, topology, which is geometric in nature, owes its origins and problems as
much to analysis as to geometry. Further the basic techniques in studying topology
are predominantly algebraic. In general, algebraic methods and symbolism pervade
all of mathematics and it is essential for anyone learning any advanced mathematics
tobefamiliarwiththeconceptsandmethodsinabstractalgebra.
This is an introductory text on abstract algebra. It grew out of courses given to
advanced undergraduates and beginning graduate students in the United States and
to mathematics students and teachers in Germany. We assume that the students are
familiarwithCalculusandwithsomelinearalgebra,primarilymatrixalgebraandthe
basic concepts of vector spaces, bases and dimensions. All other necessary material
is introduced and explained in the book. We assume however that the students have
some,butnotagreatdeal,ofmathematicalsophistication. Ourexperienceisthatthe
material in this can be completed in a full years course. We presented the material
sequentially so that polynomials and field extensions preceded an in depth look at
grouptheory. Wefeelthatastudentwhogoesthroughthematerialinthesenoteswill
attainasolidbackgroundinabstractalgebraandbeabletomoveontomoreadvanced
topics.
ThecenterpieceofthesenotesisthedevelopmentofGaloistheoryanditsimportant
applications,especiallytheinsolvabilityofthequintic. Afterintroducingthebasical-
gebraic structures, groups, rings and fields, we begin the theory of polynomials and
polynomialequationsoverfields. Wethendevelopthemainideasoffieldextensions
and adjoining elements to fields. After this we present the necessary material from
grouptheoryneededtocompleteboththeinsolvability ofthequinticandsolvability
byradicalsingeneral. Hencethemiddlepartofthebook,Chapters9through14are
concernedwithgrouptheoryincludingpermutationgroups,solvablegroups,abelian
groupsandgroupactions. Chapter14issomewhatofftothesideofthemaintheme
of the book. Here we give a brief introduction to free groups, group presentations
and combinatorial group theory. With the group theory material in hand we return
to Galois theory and study general normal and separable extensions and the funda-
mental theorem of Galois theory. Using this we present several major applications
ofthetheoryincludingsolvabilitybyradicalsandtheinsolvabilityofthequintic,the
fundamental theorem of algebra, the construction of regular n-gons and the famous
impossibilities; squaring the circling, doubling the cube and trisecting an angle. We
vi Preface
finish in a slightly different direction giving an introduction to algebraic and group
basedcryptography.
October2010 CelineCarstensen
BenjaminFine
GerhardRosenberger
Contents
Preface v
1 Groups,RingsandFields 1
1.1 AbstractAlgebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 IntegralDomainsandFields . . . . . . . . . . . . . . . . . . . . . . 4
1.4 SubringsandIdeals . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 FactorRingsandRingHomomorphisms . . . . . . . . . . . . . . . . 9
1.6 FieldsofFractions . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.7 CharacteristicandPrimeRings . . . . . . . . . . . . . . . . . . . . . 14
1.8 Groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2 MaximalandPrimeIdeals 21
2.1 MaximalandPrimeIdeals . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 PrimeIdealsandIntegralDomains . . . . . . . . . . . . . . . . . . . 22
2.3 MaximalIdealsandFields . . . . . . . . . . . . . . . . . . . . . . . 24
2.4 TheExistenceofMaximalIdeals . . . . . . . . . . . . . . . . . . . . 25
2.5 PrincipalIdealsandPrincipalIdealDomains . . . . . . . . . . . . . . 27
2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 PrimeElementsandUniqueFactorizationDomains 29
3.1 TheFundamentalTheoremofArithmetic . . . . . . . . . . . . . . . 29
3.2 PrimeElements,UnitsandIrreducibles . . . . . . . . . . . . . . . . 35
3.3 UniqueFactorizationDomains . . . . . . . . . . . . . . . . . . . . . 38
3.4 PrincipalIdealDomainsandUniqueFactorization . . . . . . . . . . . 41
3.5 EuclideanDomains . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.6 OverviewofIntegralDomains . . . . . . . . . . . . . . . . . . . . . 51
3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4 PolynomialsandPolynomialRings 53
4.1 PolynomialsandPolynomialRings . . . . . . . . . . . . . . . . . . . 53
4.2 PolynomialRingsoverFields . . . . . . . . . . . . . . . . . . . . . . 55
4.3 PolynomialRingsoverIntegralDomains . . . . . . . . . . . . . . . . 57
4.4 PolynomialRingsoverUniqueFactorizationDomains . . . . . . . . 58
4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
viii Contents
5 FieldExtensions 66
5.1 ExtensionFieldsandFiniteExtensions . . . . . . . . . . . . . . . . . 66
5.2 FiniteandAlgebraicExtensions . . . . . . . . . . . . . . . . . . . . 69
5.3 MinimalPolynomialsandSimpleExtensions . . . . . . . . . . . . . 70
5.4 AlgebraicClosures . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.5 AlgebraicandTranscendentalNumbers . . . . . . . . . . . . . . . . 75
5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6 FieldExtensionsandCompassandStraightedgeConstructions 80
6.1 GeometricConstructions . . . . . . . . . . . . . . . . . . . . . . . . 80
6.2 ConstructibleNumbersandFieldExtensions. . . . . . . . . . . . . . 80
6.3 FourClassicalConstructionProblems . . . . . . . . . . . . . . . . . 83
6.3.1 SquaringtheCircle . . . . . . . . . . . . . . . . . . . . . . . 83
6.3.2 TheDoublingoftheCube . . . . . . . . . . . . . . . . . . . 83
6.3.3 TheTrisectionofanAngle . . . . . . . . . . . . . . . . . . . 83
6.3.4 ConstructionofaRegularn-Gon . . . . . . . . . . . . . . . . 84
6.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
7 Kronecker’sTheoremandAlgebraicClosures 91
7.1 Kronecker’sTheorem . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.2 AlgebraicClosuresandAlgebraicallyClosedFields . . . . . . . . . . 94
7.3 TheFundamentalTheoremofAlgebra . . . . . . . . . . . . . . . . . 100
7.3.1 SplittingFields . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.3.2 PermutationsandSymmetricPolynomials . . . . . . . . . . . 101
7.4 TheFundamentalTheoremofAlgebra . . . . . . . . . . . . . . . . . 105
7.5 TheFundamentalTheoremofSymmetricPolynomials . . . . . . . . 109
7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
8 SplittingFieldsandNormalExtensions 113
8.1 SplittingFields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
8.2 NormalExtensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
8.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
9 Groups,SubgroupsandExamples 119
9.1 Groups,SubgroupsandIsomorphisms . . . . . . . . . . . . . . . . . 119
9.2 ExamplesofGroups . . . . . . . . . . . . . . . . . . . . . . . . . . 121
9.3 PermutationGroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
9.4 CosetsandLagrange’sTheorem . . . . . . . . . . . . . . . . . . . . 128
9.5 GeneratorsandCyclicGroups . . . . . . . . . . . . . . . . . . . . . 133
9.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
Contents ix
10 NormalSubgroups,FactorGroupsandDirectProducts 141
10.1 NormalSubgroupsandFactorGroups . . . . . . . . . . . . . . . . . 141
10.2 TheGroupIsomorphismTheorems . . . . . . . . . . . . . . . . . . . 146
10.3 DirectProductsofGroups . . . . . . . . . . . . . . . . . . . . . . . 149
10.4 FiniteAbelianGroups. . . . . . . . . . . . . . . . . . . . . . . . . . 151
10.5 SomePropertiesofFiniteGroups. . . . . . . . . . . . . . . . . . . . 156
10.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
11 SymmetricandAlternatingGroups 161
11.1 SymmetricGroupsandCycleDecomposition . . . . . . . . . . . . . 161
11.2 ParityandtheAlternatingGroups . . . . . . . . . . . . . . . . . . . 164
11.3 ConjugationinS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
n
11.4 TheSimplicityofA . . . . . . . . . . . . . . . . . . . . . . . . . . 168
n
11.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
12 SolvableGroups 171
12.1 SolvabilityandSolvableGroups . . . . . . . . . . . . . . . . . . . . 171
12.2 SolvableGroups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
12.3 TheDerivedSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
12.4 CompositionSeriesandtheJordan–HölderTheorem . . . . . . . . . 177
12.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
13 GroupsActionsandtheSylowTheorems 180
13.1 GroupActions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
13.2 ConjugacyClassesandtheClassEquation . . . . . . . . . . . . . . . 181
13.3 TheSylowTheorems . . . . . . . . . . . . . . . . . . . . . . . . . . 183
13.4 SomeApplicationsoftheSylowTheorems . . . . . . . . . . . . . . 187
13.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
14 FreeGroupsandGroupPresentations 192
14.1 GroupPresentationsandCombinatorialGroupTheory . . . . . . . . 192
14.2 FreeGroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
14.3 GroupPresentations . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
14.3.1 TheModularGroup . . . . . . . . . . . . . . . . . . . . . . 200
14.4 PresentationsofSubgroups . . . . . . . . . . . . . . . . . . . . . . . 207
14.5 GeometricInterpretation . . . . . . . . . . . . . . . . . . . . . . . . 209
14.6 PresentationsofFactorGroups . . . . . . . . . . . . . . . . . . . . . 212
14.7 GroupPresentationsandDecisionProblems . . . . . . . . . . . . . . 213
14.8 GroupAmalgams: FreeProductsandDirectProducts . . . . . . . . . 214
14.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216