数论基础

Preface

1Naturalnumbersandintegers

1.1Naturalnumbers

1.2Induction

1.3Integers

1.4Divisionwithremainder

1.5Binarynotation

1.6Diophantineequations

1.7TheDiophantuschordmethod

1.8Gaussianintegers

1.9Discussion

2TheEuclideanalgorithm

2.1Thegcdbysubtraction

2.2Thegcdbydivisionwithremainder

2.3Linearrepresentationofthegcd

2.4Primesandfactorization

2.5Consequencesofuniqueprimefactorization

2.6LinearDiophantineequations

2.7ThevectorEuclideanalgorithm

2.8Themapofrelativelyprimepairs

2.9Discussion

3Congruencearithmetic

3.1Congruencemodn

3.2Congruenceclassesandtheirarithmetic

3.3Inversesmodp

3.4Fermatslittletheorem

3.5CongruencetheoremsofWilsonandLagrange

3.6Inversesmodk

3.7QuadraticDiophantineequations

3.8Primitiveroots

3.9Existenceofprimitiveroots

3.10Discussion

4TheRSAeryptosystem

4.1Trapdoorfunctions

4.2IngredientsofRSA

4.3Exponentiationmodn

4.4RSAencryptionanddecryption

4.5Digitalsignatures

4.6Othercomputationalissues

4.7Discussion

5ThePellequation

5.1Sideanddiagonalnumbers

5.2Theequationx2-2y2=1

5.3Thegroupofsolutions

5.4ThegeneralPellequationandZ[n]

5.5Thepigeonholeargument

5.6Quadraticforms

5.7Themapofprimitivevectors

5.8Periodicityinthemapofx2-ny2

5.9Discussion

6TheGaussianintegers

6.1Zanditsnorm

6.2DivisibilityandprimesinZandZ

6.3Conjugates

6.4DivisioninZ[i]

6.5Fermatstwosquaretheorem

6.6Pythagoreantriples

6.7Primesoftheform4n+1

6.8Discussion

7Quadraticintegers

7.1Theequationy3=x2+2

7.2ThedivisionpropertyinZ[-2]

7.3ThegcdinZ[-2]

7.4Z[-3]andZ[ζ3]

7.5Rationalsolutionsofx3+y3=z3+w3

7.6Theprime-3inZ[ζ3]

7.7Fermatslasttheoremforn=3

7.8Discussion

8Thefoursquaretheorem

8.1RealmatricesandC

8.2ComplexmatricesandH

8.3Thequaternionunits

8.4Z[i,j,k]

8.5TheHurwitzintegers

8.6Conjugates

8.7Aprimedivisorproperty

8.8Proofofthefoursquaretheorem

8.9Discussion

9Quadraticreciprocity

9.1Primesx2+y2,x2+2y2,andx2+3y2

9.2Statementofquadraticreciprocity

9.3Eulerscriterion

9.4Thevalueof(2/q)

9.5Thestorysofar

9.6TheChineseremaindertheorem

9.7ThefullChineseremaindertheorem

9.8Proofofquadraticreciprocity

9.9Discussion

10Rings

10.1Theringaxioms

10.2Ringsandfields

10.3Algebraicintegers

10.4Quadraticfieldsandtheirintegers

10.5Normandunitsofquadraticfields

10.6Discussion

11Ideals

11.1Idealsandthegcd

11.2IdealsanddivisibilityinZ

11.3Principalidealdomains

11.4AnonprincipalidealofZ[-3]

11.5AnonprincipalidealofZ[-5]

11.6Idealsofimaginaryquadraticfieldsaslattices

11.7Productsandprimeideals

11.8Idealprimefactorization

11.9Discussion

12Primeideals

12.1Idealsandcongruence

12.2Primeandmaximalideals

12.3Primeidealsofimaginaryquadraticfields

12.4Conjugateideals

12.5Divisibilityandcontainment

12.6Factorizationofideals

12.7Idealclasses

12.8PrimesoftheformX2+5y2

12.9Discussion

Bibliography

Index

　　ThisbookisintendedtocomplementmyElementsofAlgebra,anditissimilarlymotivatedbytheproblemofsolvingpolynomialequations.However,itisindependentofthealgebrabook,andprobablyeasier.InElementsofAlgebrawesoughtsolutionbyradicals,andthisledtotheconceptsoffieldsandgroupsandtheirfusioninthecelebratedtheoryofGalois.Inthepresentbookweseekintegersolutions,andthisleadstotheconceptsofringsandidealswhichmergeintheequallycelebratedtheoofidealsduetoKummerandDedekind.