Math 500, Abstract Algebra

Spring 2026
Time and Place: MWF 11:00-11:50am in 136 Davenport Hall.

Instructor: William Chen
Email: wyc@illinois.edu
Office: 51 CAB (Computing Applications Building)
Office Hours: Tuesdays 3-4pm, Thursdays 1-2pm

Textbook: Dummit and Foote, Abstract Algebra, 3rd Edition. The Grainger Engineering Library has a copy on reserve for in-library use.
Supplemental text: Charles Rezk, Lecture Notes for Math 500, Fall 2022: Part 1 (Groups), Part 2 (Rings and modules), Part 3 (Fields and Galois theory)

Course Description

This is a graduate course in abstract algebra. The catalog description is:

Isomorphism theorems for groups. Group actions. Composition series. Jordan-Holder theorem. Solvable and nilpotent groups. Field extensions. Algebraic and transcendental extensions. Algebraic closures. Fundamental theorem of Galois theory, and applications. Modules over commutative rings. Structure of finitely generated modules over a principal ideal domain. Applications to finite Abelian groups and matrix canonical forms.

with more details in the official departmental syllabus.

Course Policies

Grading: Your course grade will be based on homework (40%), two in-class midterm exams (15% each), and a final exam (30%)
Weekly homework: These will typically be due on Friday before class, to be submitted via gradescope. Late homework will not be accepted, but your lowest two homework scores will be dropped. Collaboration on homework is encouraged. However, you must write up your solutions individually and understand them completely.
In-class midterms: Two 50-minute exams to be held in our usual classroom.
Final exam: TBA
Missed exams: There will be no makeup exams. If your circumstances are truly extraordinary, I may excuse you from an exam, in which case your average will be determined by your other exams and homeworks.
Cheating: No.
Disabilities: Students with disabilities who require reasonable accommodations should see me as soon as possible. In particular, any accommodation on exams must be requested at least a week in advance and will require a letter from DRES.

Schedule

Here, [DF] refers to Dummit & Foote, [R1], [R2], [R3] refer to the three parts of Rezk’s notes.
Jan 21 - Introduction. Review of groups. Sections 1.1-1.5 of [DF], Sections 1-4 of [R1] Notes
Jan 23 - Isomorphism theorems. Section 3.3 of [DF], Section 5-10 of [R1]
Jan 26 - Free groups. Section 6.3 of [DF], Sections 11-14 of [R1]
Jan 28 - Group presentations; intro to group actions. Sections 6.3 and 1.7 of [DF], Sections 15-17 of [R1]
Jan 30 - More on group actions. Section 4.1 of [DF], Sections 18-20 of [R1].
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Feb 26 - First midterm
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April 8 - Second midterm