# Teaching / Fundamental Concepts of Modern Algebra

Disclaimer: In-person lectures. No recordings, solutions, or make-up tests.

## Weekly Schedule

Asterisks indicate short weeks or weeks I am away. Exclamation marks indicate test weeks.

1 Rings, integral domains, order (MB 1.1-1.4) 1.2#1,3,5,7,8b; 1.3#1-4,7; 1.4#1,2 1.4#3-5, 1.5#6,8
2* W.o.p., induction (MB 1.5-1.8) 1.5#1,2,5(a,b, or c),11-12,14; 1.7#1,3,5,6,10,14 1.4#3-5, 1.5#6,8
3 Euclidean algorithm, FTA, valuation, congruences (MB 1.9-1.12) 1.8#1,5,8; 1.9#6,11,16
4 Maps of sets, fields (MB 1.11-1.12, 2.1-2.4) 1.11#2, 1.12 #3, 4, 2.1 #2, 6, 9 1.10#7
5 Detour on p-adic integers (Cuoco, Koblitz Ch 1) 2.3#3, 2.4#2, 3.1#7; 2.5-2.6 in Perusall
6! Polynomials (MB 3) Test 1 on Ch 1 & 2 Feb 13
7* UFDs (MB 3) 3.1#3, Prove the corollary of 3.2 theorem 4. 3. 3.2#1, 3.2#2, 3.3 # 1(c), 3.3#4 3.2#9
8 Extension fields and their automorphism groups (MB 6, 15) 3.3 #3,7; 3.5 #2,3,4,5 3.5#6 or 7
9 Groups (MB 6) 6.1#3, 6.2#3,4 Dihedral groups
March Break March Break ... March Break March ... Break
15 Death by a thousand cuts TODO TODO
11* Actions and the orbit-stabilizer theorem TODO TODO
12* Normal subgroups and quotient groups TODO TODO
13 Symmetric and alternating groups, and the platonic solids TODO TODO
14! Center, centralizers, conjugation, simplicity of $A_n$ TODO Test 2 on TBA Apr 17
15 Automorphisms (inner, outer) TODO TODO

## Specs

Instructor: Anne Dranowski (dranowsk@usc.edu)

• Office: KAP 444C or my zoom room
• Office hour windows: Tuesdays 3:45-5 in Zoom and Fridays 2:30-4:30 in person (as much as possible, please email to confirm your attendance)

TA: Kayla Orlinsky (korlinsk@usc.edu)

### Class schedule

All ~five hours of weekly classes are equally important. As much as possible we will spend class time motivating difficult concepts or elaborating on harder examples or walking through a proof together (as opposed to, say, regurgitating the details of an easy proof that can be found in the book and that you can reconstruct on your own)

• Lectures: Mondays, Wednesdays, and Fridays 1:00-1:50pm in CPA 211
• Discussion: Tuesdays 2:00-3:50pm in GFS 222
• Test 1: approx. February 13th, in class
• Test 2: approx. April 17th, in class
• Final exam: Wednesday, May 3rd, 2:00-4:00 p.m. Please review USC Policy here: https://classes.usc.edu/term-20231/finals/.

### Resources

• The Textbook: A Survey of Modern Algebra by Garrett Birkhoff & Saunders Mac Lane (4th Ed.)
• Lectures & Discussions: Some of the material we may uncover (e.g. Sylow's theorems) will be outside the scope of the book. If it is significant, I will let you know
• Math Center: The math department hosts a math help center. See their website for details
• Calendar: Session dates are summarized here https://classes.usc.edu/term-20223/calendar/

### Platforms

• Course material will be posted to the course webpage and communicated thru Blackboard announcements, therefore please check Blackboard announcements regularly.
• Homework assignments and other assessments will be administered in Gradescope, which can be accessed through Blackboard.

## Content objectives

This course will provide a detailed introduction to modern abstract algebra, which is a basic part of the language of much of modern math.

Time permitting we aim to cover parts of Chapters 1-3, 6, 9-11, and 13-15.

## Course structure

Beyond classroom time and the textbook, the course will involve:

Reading: Weekly readings will be assigned from the textbook, corresponding to the material that we are uncovering. You are expected to read the entirety of a chapter to understand it, in addition to class notes, even if we do not complete all the details in class.

Homework: There will be roughly weekly homework assignments. Late assignments will not be accepted: this course is unforgiving if you fall behind, so I do not want to encourage it. The three lowest scores will be dropped for every student.

Project: You will select one of the starred problems from the textbook and write up a detailed solution in the form of a short math article, preferably in LaTeX. Collaboration is welcome.

The final grade will consist of

• homework (13%)
• project (18%)
• two tests (18% each)
• cumulative final exam (33%)