# Math 6122: Algebra II

## Contact information:

Mike Zabrocki
Course will take place BC 323 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 4, 2019) I won't be able to be here the week of Jan 21-25 and so we will have to reschedule those classes (either reading week or at the end or some special times).
(Jan 15, 2019) I will keep a running list of exercises to solve as practice.

## Schedule (TR Winter term):

 Date Topic Remarks Jan 4 introduction, categories [DF, Appendix II] Jan 8 intro to category theory [DF, Appendix II] Jan 10 functors, free objects, R-mod [DF, Appendix II, Section 10.1, 10.2] Jan 15 free objects, R-mod, quotients [BCT, Sec 1.2, p. 70], [DF, Section 10.2, 10.3] Jan 17 Notherian <=> submodules are f.gen. [DF, Section 12.1] Jan 22 and 24 I can' be there, class to be rescheduled Jan 29 Use of Smith normal form for E.D. [DF, Section 12.1] Jan 31 Smith normal form for P.I.D., classification existence [DF, Section 12.1] Feb 5 Uniq PID module classif., f.g. ab. grps, rational can form [DF, Section 12.1 and 12.2] Feb 7 Computing the rational canonical and Jordan forms of a matrix [DF, Section 12.2 and 12.3] + Sage Feb 12 snow day Feb 14 Projective, injective and flat modules [DF, Section 10.5] Feb 19 and 21 Reading week Feb 26 Practice for midterm Feb 28 Midterm Mar 5 finish projective/injective modules, begin fields [D&F, Section 10.5, Section 13.1] Mar 7 algebraic extensions, constructible numbers [D&F, Section 13.2, 13.3] Mar 12 splitting fields and algebraic closure [D&F, Section 13.4] Mar 14 irreducible and separable polynomials [D&F, Section 13.5] Mar 19 (Kel) begin Galois theory [D&F, Section 14.1] Mar 21 perfect, separable, cyclotomic fields, ${\mathbb F}_{p^n}$ [D&F, Section 13.5 and 13.6] Mar 26 (Kel) finish Galois theory [D&F, Section 14.2] Mar 28 review for final Apr 2 (Daniel) Hilbert's Nullstellensatz [D&F, Section 15.2, 15.3] Apr 4 (Daniel) Hilbert's Nullstellensatz, problem solving Apr 9 (Oskar) Grobner bases [D&F, Section 9.6]