Description: A first graduate course in Algebra. Advanced topics in group theory: semi-direct products, dual groups, inverse limits and completions. Galois Theory, including infinite Galois xtensions. Advanced topics in ring theory: radicals, local rings, primary decomposition, the prime spectrum of a ring. Groebner Bases; elimination theory. Modules: the Cayley-Hamilton Theorem and Nakayama's lemma; localization and local properties; composition
series and the Jordan-Hölder theorem; structure theorem for finitely generated modules over a PID. Tensor products: universal properties, multilinear algebra. Introduction to categories and functors. Cross-list: MATH 563. Mutually Exclusive: Cannot register for MATH 463 if student has credit for MATH 563.