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1、以下所有内容均为个人翻译，非官方翻译版本（暂无），仅供参考。S．T．YAUCollegeStudentMathematicsContestsAlgebra,NumberTheoryandCombinatorics(seconddraft)代数、数论和组合（修改稿）LinearAlgebraAbstractvectorspaces;subspaces;dimension;matricesandlineartransformations;matrixalgebrasandgroups;determinantsandtraces;eigenvectorsandeigenvalu
2、es,characteristicandminimalpolynomials;diagonalizationandtriangularizationofoperators;invariantsubspacesandcanonicalforms;innerproductsandorthogonalbases;reductionofquadraticforms;hermitianandunitaryoperators,bilinearforms;dualspaces;adjoints.tensorproductsandtensoralgebras;线性代数抽象向量空间；子空间
3、；维度；矩阵和线性变换；矩阵代数和群；行列式和迹；特征向量和特征值；特征多项式和最小多项式；对角化和三角化；不变子空间和典范形式；内积和正交基；二次型的化简；埃尔米特算子和酉算子；双线性型；对偶空间；伴随矩阵，张量积和张量代数；IntegersandpolynomialsIntegers,Euclideanalgorithm,uniquedecomposition;congruenceandtheChineseRemaindertheorem;Quadraticreciprocity;IndeterminateEquations.Polynomials,Euclidean
4、algorithm,uniquenessdecomposition,zeros;Thefundamentaltheoremofalgebra;Polynomialsofintegercoefficients,theGausslemmaandtheEisensteincriterion;Polynomialsofseveralvariables,homogenousandsymmetricpolynomials,thefundamentaltheoremofsymmetricpolynomials.整数和多项式整数，欧几里得算法，唯一分解；同余和中国剩余定理；二次互反性；不
5、定方程.多项式，欧几里得算法，唯一分解，零点；代数基本定理；整系数多项式，高斯引理和爱森斯坦判别法；多元多项式，齐次多项式和对称多项式；对称多项式基本定理.GroupGroupsandhomomorphisms,Sylowtheorem,finitelygeneratedabeliangroups.Examples:permutationgroups,cyclicgroups,dihedralgroups,matrixgroups,simplegroups,Jordan-Holdertheorem,lineargroups(GL(n,F)anditssubgroups),
6、p-groups,solvableandnilpotentgroups,groupextensions,semi-directproducts,freegroups,amalgamatedproductsandgrouppresentations.群群和同态，西罗定理，有限生成阿贝尔群.例子：置换群、循环群、二面体群、矩阵群、单群、若尔当-赫尔德定理、线性群（GL（n，F）及其子群）、p群、可解群和幂零群，群扩张，半直积，自由群，融合积和群表示.RingBasicpropertiesofrings,units,ideals,homomorphisms,quotientri
7、ngs,primeandmaximalideals,fieldsoffractions,Euclideandomains,principalidealdomainsanduniquefactorizationdomains,polynomialandpowerseriesrings,ChineseRemainderTheorem,localringsandlocalization,Nakayama'slemma,chainconditionsandNoetherianrings,Hilbertbasistheorem,Arti
