Note:  This syllabus was approved by the GPC on May 1, 2008. 

Math 250B: Algebra (Tensor Products, Representation Theory, and Field Theory)

Prepared by Eric Babson, Greg Kuperberg, and Brian Osserman

Winter Quarter, Every Year

Lecture, 4 units

PREREQUISITES

MAT 250A or consent of instructor.

COURSE OUTLINE

Tensor products (from chapter XVI of Lang, roughly 2 weeks):

• Definition and basic properties, tensor products of algebras, and symmetric algebras.

Representation theory (from chapters XVII-XVIII of Lang, roughly 3-4 weeks):

• 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.

Algebraic field extensions (from chapters V-VI of Lang, roughly 4-5 weeks):

• 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.

READING

Lang, Serge. Algebra. 3rd edition (June 21, 2005). Springer, ISBN #038795385X. $74.95.