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1、S．T．YAUCollegeStudentMathematicsContestsAlgebra,NumberTheoryandCombinatorics(seconddraft)LinearAlgebraAbstractvectorspaces;subspaces;dimension;matricesandlineartransformations;matrixalgebrasandgroups;determinantsandtraces;eigenvectorsandeigenvalues,characteristicandminima
2、lpolynomials;diagonalizationandtriangularizationofoperators;invariantsubspacesandcanonicalforms;innerproductsandorthogonalbases;reductionofquadraticforms;hermitianandunitaryoperators,bilinearforms;dualspaces;adjoints.tensorproductsandtensoralgebras;IntegersandpolynomialsI
3、ntegers,Euclideanalgorithm,uniquedecomposition;congruenceandtheChineseRemaindertheorem;Quadraticreciprocity;IndeterminateEquations.Polynomials,Euclideanalgorithm,uniquenessdecomposition,zeros;Thefundamentaltheoremofalgebra;Polynomialsofintegercoefficients,theGausslemmaand
4、theEisensteincriterion;Polynomialsofseveralvariables,homogenousandsymmetricpolynomials,thefundamentaltheoremofsymmetricpolynomials.GroupGroupsandhomomorphisms,Sylowtheorem,finitelygeneratedabeliangroups.Examples:permutationgroups,cyclicgroups,dihedralgroups,matrixgroups,s
5、implegroups,Jordan-Holdertheorem,lineargroups(GL(n,F)anditssubgroups),p-groups,solvableandnilpotentgroups,groupextensions,semi-directproducts,freegroups,amalgamatedproductsandgrouppresentations.RingBasicpropertiesofrings,units,ideals,homomorphisms,quotientrings,primeandma
6、ximalideals,fieldsoffractions,Euclideandomains,principalidealdomainsanduniquefactorizationdomains,polynomialandpowerseriesrings,ChineseRemainderTheorem,localringsandlocalization,Nakayama'slemma,chainconditionsandNoetherianrings,Hilbertbasistheorem,Artinrings,integralringe
7、xtensions,Nullstellensatz,Dedekinddomains,algebraicsets,Spec(A).ModuleModulesandalgebraFreeandprojective;tensorproducts;irreduciblemodulesandSchur’slemma;semisimple,simpleandprimitiverings;densityandWederburntheorems;thestructureoffinitelygeneratedmodulesoverprincipalidea
8、ldomains,withapplicationtoabeliangroupsandcanonicalforms;categoriesandfunctors;complexes,injecti
