Synopsis

Topics I. Group theory Composition series and the Jordan-Hölder theorem in the context of groups with operators; simple groups and modules; solvable and nilpotent groups Group actions p-Groups and Sylow theorems Free groups; generators and relations II. Rings, fields, modules Maximal and prime ideals Comaximal ideals and Chinese Remainder Theorem Noetherian rings Principal ideal domains and unique factorization domains Polynomial rings; Hilbert's Basis Theorem; Gauss's Lemma Finite, algebraic, and primitive field extensions Presentations of modules; structure of finitely-generated modules over principal ideal domains

III. Introduction to algebraic geometry Algebraic sets and varieties Hilbert's Nullstellensatz Nilpotent elements and radical IV. Multilinear Algebra Tensor product of modules Tensor algebra of a bimodule Exterior algebra of a module over a commutative ring MATH 6310 is the first semester of a two-semester basic graduate algebra sequence. The main topics to be covered in the second semester, MATH 6320, are Galois theory, representation theory of groups and associative algebras, and an introduction to homological algebra. Prerequisites The content of a solid undergraduate course in abstract algebra, comparable to MATH 4340. Students should know the basic definitions and properties of groups, rings, modules, and homomorphisms; substructures and quotient structures; isomorphism theorems; integral domains and their fraction fields. Very little of this material will be reviewed during the course.