algebra and number theory - baker a.

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1、Algebra&NumberTheory[13/05/2003]A.BakerDepartmentofMathematics,UniversityofGlasgow.E-mailaddress:a.baker@maths.gla.ac.ukURL:http://www.maths.gla.ac.uk/»ajbContentsChapter1.BasicNumberTheory11.Thenaturalnumbers12.Theintegers33.TheEuclideanAlgorithmandthemethodofback-substitution44.Thetabularme

2、thod65.Congruences86.Primesandfactorization117.Congruencesmoduloaprime138.Finitecontinuedfractions169.Infinitecontinuedfractions1710.Diophantineequations2211.Pell’sequation23ProblemSet125Chapter2.Groupsandgroupactions291.Groups292.Permutationgroups303.Thesignofapermutation314.Thecycletypeofape

3、rmutation325.Symmetrygroups336.SubgroupsandLagrange’sTheorem357.Groupactions38ProblemSet243Chapter3.Arithmeticfunctions471.Definitionandexamplesofarithmeticfunctions472.ConvolutionandM¨obiusInversion48ProblemSet352Chapter4.Finiteandinfinitesets,cardinalityandcountability531.Finitesetsandcardina

4、lity532.Infinitesets553.Countablesets554.Powersetsandtheircardinality575.Therealnumbersareuncountable59ProblemSet460Index611CHAPTER1BasicNumberTheory1.ThenaturalnumbersThenaturalnumbers0;1;2;:::formthemostbasictypeofnumberandarisewhencountingelementsoffinitesets.Wedenotethesetofallnaturalnumber

5、sbyN0=f0;1;2;3;4;:::gandnowadaysthisisverystandardnotation.Itisperhapsworthremarkingthatsomepeopleexclude0fromthenaturalnumbersbutwewillincludeitsincetheemptyset;has0elements!WewillusethenotationZ+forthesetofallpositivenaturalnumbersZ+=fn2N:n6=0g=f1;2;3;4;:::g;0whichisalsooftendenotedN,althou

6、ghsomeauthorsalsousethistodenoteourN0.Wecanaddandmultiplynaturalnumberstoobtainnewones,i.e.,ifa;b2N0,thena+b2N0andab2N0.Ofcoursewehavethefamiliarpropertiesoftheseoperationssuchasa+b=b+a;ab=ba;a+0=a=0+a;a1=a=1a;a0=0=0a;etc:Wecanalsocomparenaturalnumbersusinginequalities.Givenx;y2N0exactlyoneof

7、thefollowingmustbetrue:x=y;xx.Inequalityistransitiveinthesensethatx