Concrete Abstract Algebra

255 Pages·2016·1.63 MB·Romanian

by  Niels Lauritzen

CONCRETE ABSTRACT ALGEBRA
FromNumberstoGro¨bnerBases
ConcreteAbstractAlgebradevelopsthetheoryofabstractalgebrafromnum-
bers to Gro¨bner bases, whilst taking in all the usual material of a traditional
introductorycourse.Inadditionthereisarichsupplyoftopicssuchascryptog-
raphy,factoringalgorithmsforintegers,quadraticresidues,finitefields,factor-
ingalgorithmsforpolynomialsandsystemsofnon-linearequations.Aspecial
feature is that Gro¨bner bases do not appear as an isolated example. They are
fullyintegratedasasubjectthatcanbetaughtsuccessfullyinanundergraduate
context.
Lauritzen’sapproachtoteachingabstractalgebraisbasedonanextensiveuse
ofexamples,applicationsandexercises.Thebasicphilosophyisthatinspiring,
non-trivial, applications and examples give motivation and ease the learning
of abstract concepts. This book is built on several years of experience teach-
ing introductory abstract algebra at Aarhus, where the emphasis on concrete
exampleshasimprovedstudentperformancesignificantly.

CONCRETE ABSTRACT ALGEBRA
From Numbers to Gro¨bner Bases
NIELS LAURITZEN
DepartmentofMathematicalSciences
UniversityofAarhus
Denmark
CAMBRIDGE UNIVERSITY PRESS
Cambridge,New York, Melbourne, Madrid, Cape Town, Singapore, Sa~o Paulo, Delhi
CambridgeUniversityPress
TheEdinburghBuilding,CambridgeCB28RU,UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521826792
(cid:1)C CambridgeUniversityPress2003
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithout
thewrittenpermissionofCambridgeUniversityPress.
Firstpublished2003
Fifth printing 2009
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcataloguerecordforthisbookisavailablefromtheBritishLibrary
LibraryofCongressCataloguinginPublicationdata
Lauritzen,Niels,1964–
Concreteabstractalgebra:fromnumberstoGro¨bnerbases/NielsLauritzen.
p. cm.
Includesbibliographicalreferencesandindex.
ISBN0521826799(hardback)–ISBN0521534100(paperback)
1.Algebra,abstract. 1.Title
QA162.L432003
512(cid:2).02–dc21 2003051248
ISBN 978-0-521-82679-9 hardback
ISBN 978-0-521-53410-9 paperback
CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracy
of URLs for external or third-partyinternetwebsitesreferredtointhispublication,
and does not guarantee that any content on suchwebsitesis,orwillremain,
accurateorappropriate. Information regarding prices, travel timetables and other
factual information given in this work are correct at the time of first printing but
Cambridge Universtiy Press does not guarantee the accuracy of such
information thereafter.
ForHelleandWilliam

Contents
Preface pagexi
Acknowledgements xiv
1 Numbers 1
1.1 Thenaturalnumbersandtheintegers 3
1.1.1 Wellorderingandmathematicalinduction 3
1.2 Divisionwithremainder 4
1.3 Congruences 5
1.3.1 Repeatedsquaring–anexample 7
1.4 Greatestcommondivisor 8
1.5 TheEuclideanalgorithm 9
1.6 TheChineseremaindertheorem 14
1.7 Euler’stheorem 17
1.8 Primenumbers 19
1.8.1 Thereareinfinitelymanyprimenumbers 20
1.8.2 Uniquefactorization 22
1.8.3 Howtocomputeϕ(n) 24
1.9 RSAexplained 24
1.9.1 Encryptionanddecryptionexponents 25
1.9.2 Findingastronomicalprimenumbers 26
1.10 Algorithmsforprimefactorization 30
1.10.1 Thebirthdayproblem 30
1.10.2 Pollard’sρ-algorithm 31
1.10.3 Pollard’s(p−1)-algorithm 33
1.10.4 TheFermat–Kraitchikalgorithm 34
1.11 Quadraticresidues 36
1.12 Exercises 41
vii
viii Contents
2 Groups 50
2.1 Definition 50
2.1.1 Groupsandcongruences 51
2.1.2 Thecompositiontable 53
2.1.3 Associativity 54
2.1.4 Thefirstnon-abeliangroup 54
2.1.5 Uniquenessofneutralandinverseelements 55
2.1.6 Multiplicationbyg ∈ G isbijective 56
2.1.7 Moreexamplesofgroups 57
2.2 Subgroupsandcosets 60
2.2.1 SubgroupsofZ 61
2.2.2 Cosets 61
2.3 Normalsubgroups 64
2.3.1 Quotientgroupsoftheintegers 66
2.3.2 Themultiplicativegroupofprimeresidueclasses 66
2.4 Grouphomomorphisms 68
2.5 Theisomorphismtheorem 71
2.6 Orderofagroupelement 72
2.7 Cyclicgroups 74
2.8 Groupsandnumbers 76
2.8.1 Euler’stheorem 76
2.8.2 Productgroups 76
2.8.3 TheChineseremaindertheorem 77
2.9 Symmetricandalternatinggroups 78
2.9.1 Cycles 79
2.9.2 Simpletranspositionsand“bubble”sort 82
2.9.3 Thealternatinggroup 85
2.9.4 Simplegroups 86
2.9.5 The15-puzzle 88
2.10 Actionsofgroups 92
2.10.1 Conjugacyclasses 98
2.10.2 Conjugacyclassesinthesymmetricgroup 98
2.10.3 Groupsoforder pr 100
2.10.4 TheSylowtheorems 101
2.11 Exercises 104
3 Rings 111
3.1 Definition 112
3.1.1 Ideals 115
3.2 Quotientrings 116
3.2.1 QuotientringsofZ 117
Contents ix
3.2.2 Primeideals 118
3.2.3 Maximalideals 118
3.3 Ringhomomorphisms 119
3.3.1 TheuniqueringhomomorphismfromZ 120
3.3.2 Freshman’sDream 121
3.4 Fieldsoffractions 123
3.5 Uniquefactorization 125
3.5.1 Divisibilityandgreatestcommondivisor 126
3.5.2 Irreducibleelements 126
3.5.3 Primeelements 127
3.5.4 Euclideandomains 130
3.5.5 Fermat’stwo-squaretheorem 132
3.5.6 TheEuclideanalgorithmstrikesagain 134
3.5.7 Primenumberscongruentto1modulo4 135
3.5.8 Fermat’slasttheorem 137
3.6 Exercises 138
4 Polynomials 143
4.1 Polynomialrings 144
4.1.1 Binomialcoefficientsmoduloaprimenumber 146
4.2 Divisionofpolynomials 147
4.3 Rootsofpolynomials 150
4.3.1 Differentiationofpolynomials 153
4.4 Cyclotomicpolynomials 154
4.5 Primitiveroots 157
4.5.1 Decimalexpansionsandprimitiveroots 159
4.5.2 Primitiverootsandpublickeycryptography 160
4.5.3 Yetanotherapplicationofcyclotomic
polynomials 160
4.6 Idealsinpolynomialrings 161
4.6.1 Polynomialringsmoduloideals 164
4.7 TheoremaAureum:thelawofquadraticreciprocity 167
4.8 Finitefields 170
4.8.1 Existenceoffinitefields 172
4.8.2 Uniquenessoffinitefields 172
4.8.3 Abeautifulidentity 173
4.9 Berlekamp’salgorithm 176
4.10 Exercises 179
5 Gro¨bnerbases 186
5.1 Polynomialsinseveralvariables 187
5.1.1 Termorderings 189