Topics_In_Number_Theory

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1、Course311:MichaelmasTerm1999PartI:TopicsinNumberTheoryD.R.WilkinsContents1TopicsinNumberTheory21.1SubgroupsoftheIntegers....................21.2GreatestCommonDivisors....................21.3TheEuclideanAlgorithm.....................31.4PrimeNumbers..

2、........................41.5TheFundamentalTheoremofArithmetic............51.6TheInnitudeofPrimes.....................61.7Congruences............................61.8TheChineseRemainderTheorem................81.9TheEulerTotientFunction................

3、...91.10TheTheoremsofFermat,WilsonandEuler..........111.11SolutionsofPolynomialCongruences..............131.12PrimitiveRoots..........................141.13QuadraticResidues........................161.14QuadraticReciprocity......................211.1

4、5TheJacobiSymbol........................2211TopicsinNumberTheory1.1SubgroupsoftheIntegersAsubsetSofthesetZofintegersisasubgroupofZif02S,x2Sandx+y2Sforallx2Sandy2S.Itiseasytoseethatanon-emptysubsetSofZisasubgroupofZifandonlyifxy2Sforallx2Sandy2S.Let

5、mbeaninteger,andletmZ=fmn:n2Zg.ThenmZ(thesetofintegermultiplesofm)isasubgroupofZ.Theorem1.1LetSbeasubgroupofZ.ThenS=mZforsomenon-negativeintegerm.ProofIfS=f0gthenS=mZwithm=0.SupposethatS6=f0g.ThenScontainsanon-zerointeger,andthereforeScontainsapositi

6、veinteger(sincex2Sforallx2S).LetmbethesmallestpositiveintegerbelongingtoS.ApositiveintegernbelongingtoScanbewrittenintheformn=qm+r,whereqisapositiveintegerandrisanintegersatisfying0r

7、hatr=0,sincemisthesmallestpositiveintegerinS.Thereforen=qm,andthusn2mZ.ItfollowsthatS=mZ,asrequired.1.2GreatestCommonDivisorsDenitionLeta1;a2;:::;arbeintegers,notallzero.Acommondivisorofa1;a2;:::;arisanintegerthatdivideseachofa1;a2;:::;arThegreatest

8、commondivisorofa1;a2;:::;aristhegreatestpositiveintegerthatdivideseachofa1;a2;:::;ar.Thegreatestcommondivisorofa1;a2;:::;arisdenotedby(a1;a2;:::;ar).Theorem1.2Leta1;a2;:::;arbeintegers,notallzero.Thenthereexistintegersu1;u2;:::;ursuchthat(a1;a2;:::;a