*Faculty*: David Harbater-
*E-mail*:`harbater AT math.upenn.edu`

*Telephone*: (215) 898-9594; Department office: (215) 898-8178-
*Office*: DRL 4E2A; Department office: DRL 4W1-6. *Office hours*: After class, by appointment, and as scheduled.

Second semester graduate algebra. Prerequisite: Math 602 or the equivalent.

Topics include:

- modules: 5 lemma, snake lemma, flat modules, projective modules, modules over a PID, localization, Nakayama's Lemma, locally free modules, Ext and Tor;
- commutative rings: Noetherian rings and modules, Hilbert Basis Theorem, Krull dimension, primary decomposition, integral extensions, integral closure, lying over theorem, Artin rings, Dedekind domains, discrete valuation rings, Krull's principal ideal theorem;
- field theory: degree of an extension, transcendence degree, algebraic closures, separable extensions, splitting fields, normal extensions, finite fields, purely inseparable extensions, Primitive Element Theorem, perfect fields;
- Galois theory: Galois extensions, fixed fields, Galois groups, Fundamental Theorem of Galois Theory, Kummer's theorem, linear independence of characters, Hilbert's Theorem 90, algebraic independence of automorphisms, Artin-Schreier theorem, normal basis theorem, geometric constructibility, solvability by radicals.

Class schedule: Mon and Fri, 1:30-3pm in DRL 4N30. There are also some classes on Wed, 1:30-3pm in DRL 4N30; details as announced.

*Textbook*: Thomas W. Hungerford, Algebra, Graduate Texts in Mathematics, vol. 73, Springer-Verlag, 1974.*Course structure*: The class meets from 1:30-3pm on Mondays and Fridays, and occasionally on Wednesdays, from January 13 to April 27. Wednesday classes either supplement or replace Monday/Friday classes, and the dates of the Wednesday classes are announced in advance. Wednesday classes include January 13 (1:30), Feb. 3 (2pm), March 2 (1:30pm), March 23 (1:30pm), April 6 (1:30), and April 27 (1:30). There will be no class on Fri., Jan. 15; Mon., Jan. 18; Fri., Feb. 12; Mon., March 7; Fri., March 11. The last day of the semester is Wed., April 27.There are weekly problem sets, which are key to the course. There is one exam, which takes place in class at the end of the semester, on Wed., April 27.

- Problem Set 1
- Problem Set 2
- Problem Set 3
- Problem Set 4
- Problem Set 5 (Hint for #2 on Problem Set 5)
- Problem Set 6
- Problem Set 7
- Problem Set 8
- Problem Set 9
- Problem Set 10
- Problem Set 11 (Optional extra problem for Problem Set 11)
- Problem Set 12