Syllabus (based on Dummit & Foote, Abstract Algebra, 3rd ed., 2004)
Prerequisites: basic knowledge of groups [DF 1.1 - 1.6, 2.1 - 2.3, 3.1 - 3.3], rings and ideals [DF 7.1 - 7.4, 9.1 - 9.2], vector spaces [DF 11.1 - 11.4]. We will rely on some of the notions introduced earlier in MTH 661 , but large segments of this course are fairly independent of MTH 661, and require only the basics above.
Fields & Galois Theory (6 weeks): Field extensions, constructibility, separable extensions, Galois theory (main theorem), cyclic and abelain extensions, computation of Galois groups, radical extensions and solvability [DF 13.1 - 13.6, 14.1 - 14.7]
Modules over PIDs (3 weeks): Basic theory, canonical forms of linear transformations, modules - tensor products and induced modules [DF 12.1 - 12.3, 10.4]
Topological groups (5 weeks): Compact groups - Haar measure and Peter-Weyl theorem, matrix groups and their Lie algebras, representations of SU(2) and SU(3), introduction to highest weight theory