丘成桐大学生数学竞赛数学专业大纲.doc

S．T．YAU College Student Mathematics Contests Algebra, Number Theory and Combinatorics (second draft) Linear Algebra Abstract vector spaces; subspaces; dimension; matrices and linear transformations; matrix algebras and groups; determinants and traces; eigenvectors and eigenvalues, characteristic and minimal polynomials; diagonalization and triangularization of operators; invariant subspaces and canonical forms; inner products and orthogonal bases; reduction of quadratic forms; hermitian and unitary operators, bilinear forms; dual spaces; adjoints. tensor products and tensor algebras; Integers and polynomials Integers, Euclidean algorithm, unique decomposition; congruence and the Chinese Remainder theorem; Quadratic reciprocity ; Indeterminate Equations. Polynomials, Euclidean algorithm, uniqueness decomposition, zeros; The fundamental theorem of algebra; Polynomials of integer coefficients, the Gauss lemma and the Eisenstein criterion; Polynomials of several variables, homogenous and symmetric polynomials, the fundamental theorem of symmetric polynomials. Group Groups and homomorphisms, Sylow theorem, finitely generated abelian groups. Examples: permutation groups, cyclic groups, dihedral groups, matrix groups, simple groups, Jordan-Holder theorem, linear groups (GL(n, F) and its subgroups), p-groups, solvable and nilpotent groups, group extensions, semi-direct products, free groups, amalgamated products and group presentations. Ring Basic properties of rings, units, ideals, homomorphisms, quotient rings, prime and maximal ideals, fields of fractions, Euclidean domains, principal ideal domains and unique factorization domains, polynomial and power series rings, Chinese Remainder Theorem, local rings and localization, Nakayamas lemma, chain conditions and Noetherian rings, Hilbert basis theorem, Artin rings, integral ring extensions, Nullstellensatz, Dedekind domains,algebraic sets, Spec(A). Module Modules and algebra Free and projective; tensor products