# Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

## MAT 250B: Algebra

Approved: 2015-11-01, Greg Kuperberg and Brian Osserman
Units/Lecture:
Winter, every year; 4 units; lecture/term paper or discussion
Suggested Textbook: (actual textbook varies by instructor; check your instructor)
Prerequisites:
250A or consent of instructor.
Suggested Schedule:

S 6.1, 6.4: Left and right modules, group representations as modules, free and projective modules (vs definition of injective), module presentations (4-5 lectures)

S 8.1, 8.3, 8.4: Classification of finitely generated modules over a PID, infinitely generated modules, Jordan canonical form, Smith normal form (3-4 lectures)

Optional: Inverse and direct limits of modules, Noetherian and Artinian modules, Jordan-Holder theorem for modules (0-4 lectures)

S 6.6: Tensor product of a left module and a right module, tensor products over commutative rings, tensor products of vector spaces with bases, definition of an algebra and tensor products of algebras, (4-6 lectures)

Optional: S 6.8: Flat modules (0-2 lectures)

S 8.5, 8.6: Bilinear forms, dual of a vector space** (Ex. 2.83), tensor, symmetric, and exterior algebras over a free module (in particular over a vector space), relation to multilinear functions (3-4 lectures)

S 2.9, 3.1-3.2: Fields as quotient rings, classification of finite fields, algebraic and Galois field extensions (including separable vs. inseparable), fundamental theorem of Galois theory, solvability by radicals, algebraic closure* (S 5.4.3), transcendence degree* (S 5.4.5) (9-12 lectures)

Optional: S 3.3: Calculation of Galois groups (0-3 lectures)

Total: 23-40 lectures

* These appear in the book in later, more advanced contexts. The instructor can follow a different treatment.

** These essential points are given as exercises. The instructor can follow a different treatment.