Math 6122: Algebra II

Contact information:

Mike Zabrocki
Course will take place TEL/DB 015 Tuesdays and Thursdays 1pm-2:15pm
Office: TEL/DB 2026
office hours: by appt.

Course description:

(short version) Chapters 10, 12, 13, 14 and Appendix II of Dummit and Foote.

(longer version) Introduction to category theory: categories, functors, basic constructions and motivating examples. Ring and module theory going beyond material covered in Math 6121 3.0: Applied Algebra; more on ideals (primes, irreducible, maximal, etc), UFD, injective and projective modules, semisimple rings and Wedderburn's theorem; Introduction to algebraic geometry: varieties, radical ideals, Hilbert’s Nullstellensatz; fields and Galois theory: field extensions, splitting fields, automorphism group of fields, Galois correspondence, Galois groups of polynomials, solving polynomials with radicals.

Course references:

Dummit and Foote, Basic Category Theory by Tom Leinster, class notes from 2017 by John Campbell

Course components:

The final grade will be base on the average of the best three of the first four of these components.
1. Project/Homework (working on an extended project or working on exercises)
2. Midterm (Writing exams).
3. Oral Presentation (Presenting some special topic or long proofs).
4. Comprehensive exam (this may also count as one of your Ph. D. requirements).

5. Participation in class (being there, asking questions, being curious, etc.) is also an important aspect of the evaluation. It may help increase any of your average above by up to 10%.

Announcements:

(Jan 7, 2020) I won't be able to be here the week of February 25-27 and so we will have to reschedule those classes.
(Jan 7, 2020) Another good reference for this class is the course web page from last year which has the schedule filled out from last year (and I hope to keep a similar pace).
(Jan 15, 2020) I will keep a running list of exercises to solve as practice.

Schedule (TR Winter term):

Date	Topic	Remarks
Jan 7	introduction, categories	[DF, Appendix II] and [BCT 1.1]
Jan 9	intro to category theory, functors	[DF, Appendix II] and [BCT Introduction, 1.1]
Jan 14	functors, problem from previous comprehensive	[DF, Appendix II] and [BCT 1.1, 1.2]
Jan 16	faithful, full, free objects, R-Mod	[DF, Section 10.1 and 10.2]
Jan 21	basis, quotients in categories and modules, direct product/sum, isomorphism theorems	[BCT, Sec 1.2, p. 70], [DF, Section 10.3]
Jan 23	CRT (exercise #16,17 DF 10.3), Noetherian modules	[DF, Section 10.3, 12.1]
Jan 28	Rough outline of procedure for SNF on E.D. (with hint how to extend to PID)	[DF Section 12.1, exercises #16-19]
Jan 30	Apply SNF to show characterization of modules over PID	[DF Section 12.1, Theorem 4,5]
Feb 4	finish CRT , uniqueness of characterization of modules over PID	[DF Section 10.3 #18, Section 12.1, Theorem 9]
Feb 6	Compute rational canonical form, Jordan canonical form	[DF Section 12.2, 12.3]
Feb 11	computer calculation of rational canonical form (using SNF), injective/projective	[DF Section 12.2, 12.3], [DF Section 10.5]
Feb 13	injective/projective, statement of TFAE for projective modules	[DF Section 10.5]
Feb 18 and 20	Reading week
Feb 25	I won't be able to be there (will need to reschedule)
Feb 27	Midterm
Mar 3	Four characterizations of projective modules	[DF Section 10.5]
Mar 5	Finish projective/injective modules, beginning fields	[DF Section 10.5], [DF Section 13.1]
Mar 10	more about projective/injective modules	[DF Section 10.5]
Mar 12	practice problem
Mar 17	fields and field extensions	[D&F Section 13.1] fields part 1
Mar 19	more fields and extensions	[D&F, Section 13.2, 13.3] fields part 2
Mar 24	splitting fields and algebraic closure	[D&F, Section 13.4] fields part 3
Mar 26	irreducible and separable polynomials	[D&F, Section 13.5]
Apr 2	(Jordan) Galois theory	[D&F, Section 14.1]
Apr 7	finish Galois theory
Apr 14	(Kailun) Grobner bases