Mondays and Wednesdays 12-1:15 SC Hall E

This class is an introduction to abstract algebra. The focus of this semester will be groups, rings, and modules.

**Instructor:** Brooke Ullery (bullery@math, office SC 503)

**Teaching fellow:** Zijian Yao (zyao@math)

**Course assistants:** CJ Dowd (cjdowd@college), Charlie O'Mara (comara@college), Fan Zhou (fanzhou@college), Matthew Hase-Liu (matthewhaseliu@college)

**Office hours:**

- Monday 1:15-2:20 PM (Brooke, SC 503)
- Monday 8-10 PM (CJ and Matthew, Math Night in Leverett dining hall)
- Tuesday 1-2 PM (Brooke, SC 503)
- Tuesday 1:30-2:45 PM (Fan, SC 309A)
- Wednesday 4:15-6:15 PM (Charlie, Kirkland junior common room)

**Section:** Sundays at 7 PM in SC Hall A, run by Zijian.

**Text:** Abstract Algebra, by Dummit and Foote.

**Homework:** Problem sets will be assigned weekly (usually due Wednesdays). You should submit them on canvas. You can either type your solutions using latex, or very neatly handwrite your solutions, and scan them. I encourage you to work on the problems together, but you must turn in your own solutions and list the names of your collaborators. You may not use solutions found anywhere online. If you consult any outside source, make sure you clearly cite it.

**Exams:** There will be one in-class midterm on October 30. You can bring one 3 inch by 5 inch card or piece of paper (two-sided) with notes on it. Here's a midterm study guide. Here are the midterm solutions. There will be a take-home final from December 4-7 or 7-10 (your choice).

**Grading:**
Homework: 60%, Exams: 20% each

**Problem set 1:** pdf file, tex file, solutions

**Problem set 2:** pdf file, tex file, solutions

**Problem set 3:** pdf file, tex file, solutions

**Problem set 4:** pdf file, tex file, solutions

**Problem set 5:** pdf file, tex file, solutions

**Problem set 6:** pdf file, tex file, solutions

**Problem set 7:** pdf file, tex file, solutions

**Problem set 8:** pdf file, tex file, solutions

**Problem set 9:** pdf file, tex file, solutions

**Problem set 10:** pdf file, tex file, solutions

**Problem set 11 (extra credit):** pdf file, tex file, due December 3

You can find the lecture notes from class here. I'll post each section after we've covered it, so there may be some notes covering multiple days. Warning: There will inevitably be typos in the notes!

Section 1: Binary operations

Section 2: Groups

Section 3: Dihedral groups

Section 4: Symmetric groups

Section 5: Group homomorphisms

Section 6: Group actions

Section 7: The subgroup criterion

Section 8: Centralizers, normalizers, stabilizers

Section 9: Cyclic groups and cyclic subgroups

Section 10: Subgroups generated by subsets of a group

Section 11: Quotient groups and normal subgroups

Section 12: Cosets and orders of subgroups

Section 13: The isomorphism theorems

Section 14: The alternating group

Section 15: Group actions and permutation representations

Section 16: Groups acting on themselves by left multiplication and Cayley's theorem

Section 17: Groups acting on themselves by conjugation and the class equation

Section 18: Sylow's Theorem

Section 19: Rings

Section 20: Polynomial rings

Section 21: Ring homomorphisms and quotient rings

Section 22: Properties of ideals

Section 23: Euclidean domains

Section 24: Principal ideal domains

Section 25: Unique factorization domains

Section 26: Polynomial rings revisited

Section 27: Fields of fractions and Gauss' Lemma

Section 28: Irreducibility criteria