## Math 120: Groups and Rings

In Fall 2015 I taught Math 120 at Stanford University. The course assistant was Niccolò Ronchetti. For questions about the material and class discussions, we used the Math 120 Piazza page. Office hours: Church Monday 4–5:30 and Thursday 4–5 in 383-Y; Ronchetti Tuesday 6–7:30 and Friday 6–7:30 in 381-M.

Homework:

The take-home midterm exam, and the take-home final exam.
[Note to teachers: both exams were on the easy side, quite possibly too easy for a take-home exam. Q5 on the final could be "brute-forced" in terms of entries of C, which made it easier than intended.]

Material covered:

• Week 1: 1.1, 1.3, 1.4, 1.6 (9/21: 1.1; 9/23: 1.4, begin 1.6; 9/25: 1.6, 1.3, begin 1.7)
• Online assignment #1 (on Coursework, due 9/27): read 1.7, 2.1, 2.2
• Week 2: 1.7, 2.3, 2.4, 3.1 (9/28: 1.7, 9/30: 3.1, 10/2: 2.4, 2.3)
• Online assignment #2 (due 10/4): read 3.2 through Corollary 10, 3.3, 4.1, 4.2
• Week 3: 3.3, 3.2, 6.3, 4.3 (10/5: 3.2, 3.3, 10/7: 6.3, 10/9: start 4.3)
• Online assignment #3 (due 10/11): read 4.3 through Proposition 10, 4.2, statement of Sylow theorem in 4.5
• Week 4: 4.3, 4.2, 4.5
• Online assignment #4 (due 10/18): read 7.1 and 7.2 carefully, try to understand these examples of rings (OK to skip "Quadratic Integer Rings" on p229 and group rings on p236)
• Week 5: 7.1, 7.2, 7.3
• Online assignment #5 (due 10/25): 7.4, 7.6, skim 7.5 without getting overwhelmed by details
• Week 6: 7.4, 7.6, 8.3
• Online assignment #6 (due 11/1): 8.1 through Theorem 4, 8.2 through Corollary 8, 8.3 through Corollary 15
• Week 7: 8.3, 8.2, 8.1
• Online assignment #7 (due 11/8): 9.3, 10.1 through p339 (you can look at 340-341 if you want)
• Week 8: 9.3, modules, isomorphisms of modules, free modules and basis
• Online assignment #8 (due 11/15): §2.1–2.5 of Reid's "Undergraduate Commutative Algebra" (skip first two full paragraphs on p38)
• Week 9: structure theorem for f.g. modules over a PID; applications: linear algebra, classification of f.g. abelian groups
• Week 10: continue with modules over a PID: rational canonical form, torsion-free implies free.
• Last lecture: semisimple rings and representation theory of finite groups in characteristic 0 (number and dimension of simple representations).

WIM paper topics:
1) alternating group An is simple for n≥5, or