# 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.

**Approved:**2015-11-01, Greg Kuperberg and Brian Osserman

**Units/Lecture:**

**Suggested Textbook:**(actual textbook varies by instructor; check your instructor)

https://www.amazon.com/Advanced-Algebra-Graduate-Studies-Mathematics/dp/B00N4J2BP6/

**Prerequisites:**

**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.

**Additional Notes:**

Rotman is a thick book with easy sections that can be accelerated and difficult sections that should not be covered completely. Proofs of some of the harder results listed in the syllabus can be taken as optional.

Rotman bases his presentation of homological algebra on derived functors. However, one can skip this machinery in subsequent sections, particularly by assuming existence results. For instance, axioms characterizing Ext and Tor are stated without reference to derived functors.