Basic Topics
Course material- Mathematics 110ABC, 115AB, and 210A.
Topics- Group Theory: elementary theory dealing with properties of finite groups, Lagrange's theorem, permutation groups, abstract groups including automorphisms and homomorphisms, normal subgroups and factors groups, the homomorphism theorems, and Sylow theorems.
Ring Theory: elementary properties of rings, ideals and homomorphisms; integral domains and factorization; polynomials including factorization, existence and type of roots; Application to elementary number theory.
Field Theory: the rational, real and complex number fields, finite fields, quotient fields, field extensions and Galois theory, geometric constructions and solvability by radicals.
Linear Algebra: abstract vector spaces; subspaces; dimension theory, inner products and orthogonal bases, matrices and linear transformations, matrix algebras and groups, determinants and traces, eigenvectors and eigenvalues, canonical forms, reduction of quadratic forms, bilinear forms.
More Advanced Topics
Course material- Mathematics 110ABC, 115AB and 210 ABC.
Topics- Group Theory: through the Sylow, Jordan-Holder and Krull-Schmidt theorems; permutation representations; nilpotent, solvable and simple groups; matrix groups; and free groups.
Ring Theory: factorization and localization, chain conditions, Hilbert basis theorem, integral extensions, primary decompositions and the Hilbert Nullstellensatz.
Module Theory: free and projective modules, tensor products, irreducible modules and Schur's lemma; semisimple, simple and primitive rings; density and Wederburn theorems; the structure of finitely generated modules over principal ideal domains and application to abelian groups and canonical forms.
Field Theory: transcendence bases, separable and inseparable extensions; Galois theory, finite fields, cyclic and cyclotomic extensions.
Linear Theory: (In addition to MA level material) dual spaces, tensor products and algebras, and linear groups.
