MATH 418 - Abstract Algebra II (Spring 2024)

| 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.

General Information

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 2:00pm-2:50pm, CAB 69B
More Details	Syllabus | Gradescope | If you can't access the Gradescope course, email me

Homework

Homework assignments are weekly, due at 9am on Wednesdays via Gradescope. Using LaTeX for your homework is encouraged, and you will receive 2 bonus points for each homework you typeset using LaTeX.

• Homework 1 (TeX file) -- due Wed. 1/24 at 9am (Solutions)
• Homework 2 (TeX file) -- due Wed. 1/31 at 9am (Solutions)
• Homework 3 (TeX file) -- due Wed. 2/7 at 9am (Solutions)
• Homework 4 (TeX file) -- due Wed. 2/21 at 9am (Solutions)
• Homework 5 (TeX file) -- due Wed. 2/28 at 9am (Solutions)
• Homework 6 (TeX file) -- due Wed. 3/6 at 9am (Solutions)
• Homework 7 (TeX file) -- due Fri. 3/29 at 9am (Solutions)
• Homework 8 (TeX file) -- due Wed. 4/10 at 9am (Solutions)
• Homework 9 (TeX file) -- due Wed. 4/24 at 9am (Solutions)
• Homework 10 (TeX file) -- due Wed. 4/31 at 9am (Solutions)

Lecture Notes

Lecture notes will be posted here

• Lecture 1 (1/17): Syllabus, course overview
• Lecture 2 (1/19): Euclidean domains
• Lecture 3 (1/22): Principal ideal domains
• Lecture 4 (1/24): Unique factorization domains
• Lecture 5 (1/26): Gauss' Lemma and unique factorization in polynomial rings
• Lecture 6 (1/29): Unique factorization in polynomial rings and irreducibility criteria
• Lecture 7 (1/31): Irreducibility criteria (cont.) and field extensions
• Lecture 8 (2/2): Field extensions (cont.)
• Lecture 9 (2/5): Field extensions (cont.), Algebraic extensions
• Lecture 10 (2/7): Algebraic extensions (cont.)
• Lecture 11 (2/9): Algebraic extensions (cont.)
• Lecture 12 (2/12): Algebraic extensions (cont.), straightedge and compass construction
• Lecture 13 (2/14): Midterm 1 Review
• Lecture 14 (2/16): Straightedge and compass construction (cont.)
• Lecture 15 (2/19): Splitting fields
• Lecture 16 (2/21): Algebraic closures
• Lecture 17 (2/23): Fundamental Theorem of Algebra and separability
• Lecture 18 (2/26): Separability (cont.)
• Lecture 19 (2/28): Finite fields and cyclotomic fields
• Lecture 20 (3/1): Cyclotomic fields (cont.) and intro to Galois theory
• Lecture 21 (3/4): Automorphism groups and fixed fields
• Lecture 22 (3/6): Automorphism groups (cont.)
• Lecture 23 (3/18): Primitive element theorem
• Lecture 24 (3/20): Midterm 2 Review
• Lecture 25 (3/22): Finite fields, fixed fields
• Lecture 26 (3/25): Towards the Fundamental Theorem of Galois Theory
• Lecture 27 (3/27): Fundamental Theorem of Galois Theory
• Lecture 28 (3/29): Cyclotomic Galois groups and constructibility of n-gon
• Lecture 29 (4/1): Galois groups of polynomials
• Lecture 30 (4/3): Galois groups of polynomials (cont.)
• Lecture 31 (4/5): Cubic formula
• Lecture 32 (4/10): Solvability by radicals
• Lecture 33 (4/12): Solvability by radicals (cont.)
• Lecture 34 (4/15): Intro to algebraic geometry
• Lecture 35 (4/17): Midterm 3 Review
• Lecture 36 (4/19): Hilbert's Nullstellensatz
• Lecture 37 (4/22): Prime ideals and irreducible varieties
• Lecture 38 (4/24): Proof of the Nullstellensatz
• Lecture 39 (4/26): Projective space
• Lecture 40 (4/29): Projective varieties
• Lecture 41 (5/1): Final Exam Review

Important Course Events

Midterm Exam 1: Thursday, 2/15, 7:00pm-8:30pm, Loomis Laboratory 144 (Solutions)
Midterm Exam 2: Thursday, 3/21, 7:00pm-8:30pm, Loomis Laboratory 144 (Solutions)
Midterm Exam 3: Thursday, 4/18, 7:00pm-8:30pm, Loomis Laboratory 144 (Solutions)
Final Exam: Tuesday 5/7 -- 8:00am-11:00am, 1047 Sidney Lu Mechanical Engineering Building (Solutions)

Problem Sessions: Tuesdays, 3:00pm-4:20pm in 2200 Sidney Lu Mechanical Engineering Building

• Solutions to Practice Problems 1
• Solutions to Practice Problems 2
• Solutions to Practice Problems 3
• Solutions to Practice Problems for Final Exam
• Gradeline calculator -- only works in Google Sheets

Background and Other Resources

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.

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

• David Cox, John Little, and Donal O'Shea, Ideals, Varieties, and Algorithms

Helpful Links

• Overleaf, a cloud-based LaTeX editor
• Detexify, a web app for identifying the LaTeX command(s) for various symbols
• Guidelines for Good Mathematical Writing, by Francis Su

Bonus Links

Useful, relevant, and/or fun. Feel free to send me anything interesting you run across.