MATH 418 - Abstract Algebra II (Spring 2025)
| General Information |
Homework |
Lecture Notes |
Other Resources |
This is a second course in abstract algebra. We will cover three main topics:
- Ring theory: Euclidean domains, principal ideal domains, unique factorization domains, polynomial rings and irreducibility
- Field theory: (algebraic) field extensions, constructibility, and Galois theory
- Algebraic geometry: varieties, Hilbert's Nullstellensatz, and possibly more (depending on time)
Galois theory is a subtle, yet beautiful subject which characterizes solutions to polynomial equations through connections between groups and fields. All but the last portion of the course will be spent building up to and proving fundamental results in this area, with applications to problems first considered thousands of years ago.
| Lecture | MWF 1:00pm-1:50pm
1047 Sidney Lu Mechanical Engineering Building |
| Textbook | David Dummit & Richard Foote
Abstract Algebra, 3rd Edition
Chapters 8, 9, 13-15 (roughly) |
| Instructor | Andrew Hardt
Office: CAB 69B
Email: ahardt@illinois.edu
Office Hours: MF 12:00pm-12:50pm, CAB 69B |
| More Details | Syllabus | Gradescope | If you can't access the Gradescope course, email me |
Homework assignments are weekly, due at 9am on Wednesdays via Gradescope. Using LaTeX for your homework is encouraged, and you will receive 1 bonus point for each homework you typeset using LaTeX.
- Lecture 1 (1/22): Syllabus, course overview
- Lecture 2 (1/24): Euclidean domains
- Lecture 3 (1/27): Principal ideal domains
- Lecture 4 (1/29): Unique factorization domains
- Lecture 5 (1/31): Gauss' Lemma
- Lecture 6 (2/3): Unique factorization in polynomial rings
- Lecture 7 (2/5): Irreducibility criteria
- Lecture 8 (2/7): Field extensions
- Lecture 9 (2/10): Field extensions (cont.)
- Lecture 10 (2/12): Algebraic extensions
- Lecture 11 (2/14): Tower law and composite extensions
- Lecture 12 (2/17): Composite extensions (cont.), straightedge and compass construction
- Lecture 13 (2/19): Midterm 1 Review
- Lecture 14 (2/21): Straightedge and compass construction (cont.)
- Lecture 15 (2/24): Splitting fields
- Lecture 16 (2/26): Algebraic closures, Fundamental Theorem of Algebra
- Lecture 17 (3/3): Separability
- Lecture 18 (3/5): Separability (cont.)
- Lecture 19 (3/7): Finite fields and cyclotomic fields
- Lecture 20 (3/10): Cyclotomic fields (cont.) and intro to Galois theory
- Lecture 21 (3/12): Automorphism groups and fixed fields
- Lecture 22 (3/14): Automorphism groups (cont.)
- Lecture 23 (3/24): Primitive element theorem
- Lecture 24 (3/26): Midterm 2 Review
- Lecture 25 (3/28): Finite fields, Galois conjugates
- Lecture 26 (3/31): Towards the Fundamental Theorem of Galois Theory
- Lecture 27 (4/2): Fundamental Theorem of Galois Theory
- Lecture 28 (4/4): Cyclotomic Galois groups and constructibility of n-gon
- Lecture 29 (4/7): Symmetric functions, and the discriminant
- Lecture 30 (4/9): Galois groups of polynomials
- Lecture 31 (4/11): Cubic formula
- Lecture 32 (4/14): Solvability by radicals
- Lecture 33 (4/16): Solvability by radicals (cont.)
- Lecture 34 (4/18): Intro to algebraic geometry
- Lecture 35 (4/21): Hilbert's Nullstellensatz, weak and strong forms
- Lecture 36 (4/23): Midterm 3 Review
Important Course Events
Midterm Exam 1: Wednesday, 2/19, 7:00pm-8:30pm, Sidney Lu 1043 (Solutions)
Midterm Exam 2: Wednesday, 3/26, 7:00pm-8:30pm, Sidney Lu 1043 (Solutions)
Midterm Exam 3: Wednesday, 4/23, 7:00pm-8:30pm, Sidney Lu 1043
Final Exam: Tuesday 5/13 -- 8:00am-11:00am, Sidney Lu 1047
Problem Sessions: Tuesdays, 3:00pm-4:20pm, Loomis Lab 143
Background
Most of the necessary background for this course was covered in Math 417: Abstract Algebra I. 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. In particular, Chapter 7 is an important prerequisite. If you haven't taken Math 416, it would be a good idea to read through Section 11.1 as well.
Galois Theory Resources
- Emil Artin, Galois Theory (open access)
- Serge Lang, Algebra, 3rd Edition, Chapters 5-6 (available if logged in through UIUC)
- Bourbaki, Algebra, Chapters IV-V (different, much more sophisticated perspective)
- Some more sources are mentioned here and here.
Algebraic Geometry Resources
2024 Course
Helpful Links
Bonus Links
Useful, relevant, and/or fun. Feel free to send me anything interesting you run across.