Meetings: Every weekday from September 8 to September 18 from 2:00pm - 4:15pm in MSB 2112.
Instructor: Babson
Email: b a b s o n@math.ucdavis.edu
Website:
Office: 2109 MSB

Topics: There are at least three official lists of topics which are relevant. The more recent and hence more correct are the newly submitted syllabi for this course:

Course Description: Problem-solving in graduate algebra: groups, rings, modules, matrices, tensor products, representations, Galois theory, ring extensions, commutative algebra and homological algebra.
Expanded Course Description
1. Summary of Course Contents: The course is intended as a workshop on solving problems in Algebra. The topics covered in the class include the following.
1) Groups. Solvable groups. Semidirect products. Abelian groups. Sylow's theorems. Free groups. Groups presentations.
2) Rings. Chinese remainder theorem. Polynomial and group rings. Localization. Ideals. Principal ideal domains. Unique factorization domains. Hilbert's basis theorem.
3) Modules. Free modules. Projective modules. Noetherian and Artinian properties. Modules over a principal ideal domain.
4) Matrices. Modules over a polynomial ring. Jordan form. Bilinear forms.
5) Tensor Products. Universal properties. Tensor products of modules and algebras. Symmetric and alternating algebras.
6) Representations. Semisimple rings. Characters and orthogonality. Representations of finite groups (including Abelian and symmetric ones).
7) Galois Theory. Algebraic, normal, separable, solvable and radical field extensions. Finite fields. Algebraic closures. Splitting fields. The fundamental theorem of Galois theory. Solvability by radicals.
8) Ring extensions. Integral extensions. Noether's normalization theorem.
9) Commutative Algebra. Nullstellensatz. Primary decompositions. Dedekind domains.
10) Homological Algebra. Chain complexes. Homology. Exactness of functors. Long exact sequences. Injective, projective and flat resolutions.

Math 250 syllabus
Textbook: Lang, Algebra
250A: Groups, rings, and modules
I. Groups (from chapter I of Lang)
Rapid review of basic group theory, followed by topics including: solvable groups, the Holder program, Sylow theorems, abelian groups, semidirect products, profinite groups, and free groups, presentations, and related constructions.
II. Rings (from chapters II and IV of Lang)
Rapid review of basic ring theory, followed by topics including: the Chinese Remainder Theorem, polynomial and group rings, localization, principal ideal rings, and the Hilbert basis theorem.
III. Modules (from chapter III of Lang)
Basic definitions and constructions, free and projective modules, bilinear forms, and structure of modules over principal ideal rings.
250B: Field theory, tensor products, and representation theory
V. Tensor products (from chapter XVI of Lang) Definition and basic properties, tensor products of algebras, and symmetric algebras.
VI. Representation theory (from chapters XVII and XVIII of Lang)
Semisimplicity. Basic definitions of representations and characters, and structural results, 1-dimensional representations and representations of finite abelian groups, and class functions and orthogonality of characters.
VII. Algebraic field extensions (from chapters V and VI of Lang)
Rapid review of finite and algebraic field extensions, followed by topics including: algebraic closures, normal extensions, separable and inseparable extensions, finite fields, the fundamental theorem of Galois theory, roots of unity, solvable and radical extensions, and solvability by radicals, and infinite Galois theory.

Problems: The best way to prepare for an exam is to practice writing up solutions to appropriate problems. Here are various lists of problems to try: