## 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:**

Ch. 1: Rapid review of undergraduate group theory (1-2 lectures)

Ch. 4: Classification of finite abelian groups, Sylow theorems,

Jordan-Holder theorem, semidirect products* (S 9.2), structure of

nilpotent groups, free groups and group presentations (8-10 lectures)

*Optional: Central extensions* (S 9.8), Nielsen-Schreier theorem (0-3 lectures)*

S 2.1-2.8: Rapid review of undergraduate ring theory and linear

algebra, eigenvalues and characteristic polynomials* (S 8.2)

(3-4 lectures)

S 5.1-5.4.1, 5.5.4: Prime and maximal ideals, unique factorization

domains, generalized Chinese remainder theorem, Noetherian rings (vs

definition of Artinian* (S 7.1)), quotients of polynomial rings as

ring presentations***, Hilbert basis theorem, local rings** (Ex. 5.13),

localization of rings* (S 10.2) (7-10 lectures)

*Optional: S 5.5.4: primary decomposition, Noether-Lasker theorem (0-2 lectures)*

S 5.6, 5.7: Multivariate polynomial division, Grobner bases (4-6 lectures)

Total: 23-37 lectures

*** This is not explicitly pointed out in Rotman.

**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.