MATH 121 - Galois Theory (Winter 2023)

Galois theory is a subtle, yet beautiful subject which characterizes solutions to polynomial equations through connections between groups and fields. It was the brainchild of a troubled teenager, Évariste Galois, grieving his father's death and his college rejection, who was repeatedly arrested for his political activism. Galois sadly died in a duel, and his work wasn't fully appreciated until many decades later, but has since become an essential part of mathematics, with connections to advanced topics in number theory, algebraic geometry, and elsewhere.

General Information

Lecture	MWF 9:30am-10:20am Building 200, Rm 205
Textbook	David Dummit & Richard Foote Abstract Algebra, 3rd Edition (primarily chapters 13 and 14)
Instructor	Andy Hardt Office: Building 380, Rm 382C Email: hardt@stanford.edu Office Hours: MWF 10:30am-11:30am
Course Assistant	Yuefeng Song Office: Building 380, Rm 381L Email: songyf@stanford.edu Office Hours: M 1:00pm-3:00pm, TuTh 9:00am-10:00am
More Details	Canvas page \| Syllabus \| (for Gradescope link, see Canvas)

Homework

Homework assignments are weekly, due at noon on Tuesdays via Gradescope. All problems are from Dummit and Foote, unless stated otherwise. Using LaTeX for your homework is encouraged (and good practice!), but not required.

Homework 1 (due Tues. 1/17 at noon): 13.1: #1, 2, 3, 4, 6, 7 (Solutions)
Homework 2 (due Tues. 1/24 at noon): 13.2: #1, 4, 5, 7, 12, 15; 13.3: #2, 4 (Solutions)
Homework 3 (due Tues. 1/31 at noon): 9.4: #2d, 10, 12; 13.4: #1, 2, 3, 5, 6 (Solutions)
Homework 4 (due Tues. 2/7 at noon): 13.5: #3, 6; 13.6: #2, 3, 7; 14.1: #3, 5, 10 (Solutions)
Homework 5 (due Tues. 2/21 at noon): 14.2: #6, 7, 14, 19, 23; 14.3 #3, 4, 8 (Solutions)
Homework 6 (due Tues. 3/7 at noon): 14.4: #1, 3; 14.5 #3, 7, 10; 14.6 #2a, 22, plus this problem (Solutions)
Homework 7 (due Tues. 3/14 at noon): 14.6: #8, 10; 14.7 #1, 4, 9, 17 (Solutions)

Project

This project entails computing the Galois correspondence and associated data for different polynomials. Due via Gradescope on Friday, March 3rd, 2023 at noon.

Project Description

Lecture Notes

Lecture 1 (1/9): Course Overview
Lecture 2 (1/11): Section 13.1
Lecture 3 (1/13): Section 13.1 (cont.), Section 13.2
Lecture 4 (1/18): Section 13.2 (cont.)
Lecture 5 (1/20): Section 13.3
Lecture 6 (1/23): Section 9.4
Lecture 7 (1/25): Section 13.4
Lecture 8 (1/27): Section 13.4 (cont.), Section 13.5
Lecture 9 (1/30): Section 13.5 (cont.), Section 13.6
Lecture 10 (2/1): Section 13.6 (cont.), Section 14.1
Lecture 11 (2/3): Section 14.1 (cont.) (Extra example)
Lecture 12 (2/6): Section 14.2
Lecture 13 (2/8): Midterm Exam Review
Lecture 14 (2/10): Section 14.2 (cont.)
Lecture 15 (2/13): Section 14.2 (cont.)
Lecture 16 (2/15): Section 14.2 (cont.) (Extra example)
Lecture 17 (2/17): Section 14.3
Lecture 18 (2/22): Section 14.4
Lecture 19 (2/24): Section 14.4 (cont.), Section 14.5
Lecture 20 (2/27): Section 14.5 (cont.) (Complex constructability)
Lecture 21 (3/1): Section 14.5 (cont.), Section 14.6
Lecture 22 (3/3): Section 14.6 (cont.)
Lecture 23 (3/6): Section 14.6 (cont.)
Lecture 24 (3/8): Section 14.7
Lecture 25 (3/10): Section 14.7 (cont.)
Lecture 26 (3/13): Section 14.8
Lecture 27 (3/15): Section 14.9
Lecture 28 (3/17): Final Exam Review

Schedule

Midterm Exam: Wednesday, February 8th – 7:00PM – 9:00PM (Room 200-205)
Final Exam: Thursday, March 23rd – 8:30AM – 11:30AM (Room 200-205)

Background and Other Resources

Background

Most of the necessary background for this course was covered in Math 120: Groups and Rings. Having a working knowledge of topics from that course will be invaluable in this one. All relevant material can be found in earlier sections of Dummit & Foote. We will cover/review some of the most important ideas, but here are some particularly relevant topics:

Basic concepts for groups, rings, fields, vector spaces
Group/ring homomorphisms and isomorphisms
Polynomial rings and irreducibility (Dummit & Foote chapter 9; we will cover some of this material)

Other Galois Theory Resources

Emil Artin, Galois Theory (open access)
Serge Lang, Algebra, 3rd Edition, Chapters 5-6 (available through Stanford library website)
Bourbaki, Algebra, Chapters IV-V (different, much more sophisticated perspective)
Some more sources are mentioned here and here.

Bonus Links

Relevant to Galois theory, and (perhaps) fun. Feel free to send me anything interesting you run across.