内容简介
ThisbookisintendedtocomplementmyElementsofAlgebra,anditissimilarlymotivatedbytheproblemofsolvingpolynomialequations.However,itisindependentofthealgebrabook,andprobablyeasier.InElementsofAlgebrawesoughtsolutionbyradicals,andthisledtotheconceptsoffieldsandgroupsandtheirfusioninthecelebratedtheoryofGalois.Inthepresentbookweseekintegersolutions,andthisleadstotheconceptsofringsandidealswhichmergeintheequallycelebratedtheoryofidealsduetoKummerandDedekind.
Thebookisbasedontwoshortcourses(about20lectureseach)givenatMonashUniversityinrecentyears;oneonelementarynumbertheoryandoneonringtheorywithapplicationstoalgebraicnumbertheory.Thustheamountofmaterialissuitableforaone-semestercourse,withsomevariationpossiblethroughomissionoftheoptionalstarredsections.Aslower~pacedcoursecouldstopattheendofChapter9,atwhichpointmostofthestandardresultshavebeencovered,fromEuclid'stheoremthatthereareinfinitelvmanyprimestoquadraticreciprocity.目录
Preface
1Naturalnumbersandintegers
1.1Naturalnumbers
1.2Induction
1.3Integers
1.4Divisionwithremainder
1.5Binarynotation
1.6Diophantineequations
1.7TheDiophantuschordmethod
1.8Gaussianintegers
1.9Doscission
2TheEuclideanalgorithm
2.1Thegcdbysubtraction
2.2Thegcdbydivisionwithremainder
2.3Linearrepresentationofthegod
2.4Primesandfactorization
2.5Consequencesofuniqueprimefactorization
2.6LinearDiophantineequations
2.7ThevectorEudideanalgorithm
2.8Themapofrelativelyprimepairs
2.9Doscission
3Congruencearithmetic
3.1Congruencemodn
3.2Congruenceclassesandtheirarithmetic
3.3Inversesmodp
3.4Fermat'slittletheorem
3.5CongruencetheoremsofWilsonandLagrange
3.6Inversesmodk
3.7QuadraticDiophantineequations
3.8Primitiveroots
3.9Existenceofprimitiveroots
3.10Discussion
4TheRSAcryptosystem
4.1Trapdoorfunctions
4.2IngredientsofRSA
4.3Exponentiationmodn
4.4RSAencryptionanddecryption
4.5Digitalsignatures
4.60thercomputationalissues
4.7Discussion
5ThePellequation
6TheGaussianintegers
7Quadraticintegers_
8Thefoursquaretheorem
9Quadraticreciprocity
10Rings
11Ideals
12Primeideals
Bibliography
Index