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

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