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] [Notes]
Jan 26 - Free groups. Section 6.3 of [DF], Sections 11-14 of [R1] [Notes]
Jan 28 - Group presentations; intro to group actions. Sections 6.3 and 1.7 of [DF], Sections 15-17 of [R1] [Notes]
Jan 30 - More on group actions, orbit-stabilizer theorem. Section 4.1 of [DF], Sections 18-20 of [R1] [Notes][Problem Set 1] due [pset1 solutions]
Feb 2 - Applications of group actions, conjugacy classes. Sections 4.2-3 of [DF], Sections 21-25 of [R1] [Notes]
Feb 4 - Automorphism groups of groups. Sections 4.4-5 of [DF], Sections 26-29 of [R1] [Notes]
Feb 6 - Sylow theorems. Section 4.5 of [DF], Sections 29-34 of [R1] [Notes][Problem Set 2] due [pset2 solutions]
Feb 9 - Finitely generated groups and the ascending chain condition. Section 36-38 of [R1] [Notes]
Feb 11 - Torsion in abelian groups and direct products. Section 5.1 of [DF], Sections 38-41 of [R1] [Notes]
Feb 13 - Classification of f.g. abelian groups. Group extensions. Section 5.2 of [DF], Sections 43-45 of [R1]. [Notes][Problem Set 3] due. [pset3 solutions]
Feb 16 - Composition series and the Jordan-Holder theorem. Sections 5.5 and 3.4 of [DF] and Sections 46-48 of [R1]. [Notes]
Feb 18 - Solvable and nilpotent groups. Section 6.1 of [DF] and 49-51 of [R1] [Notes]
Feb 20 - Review of rings. Sections 7.1 and 7.3 of [DF] and 1-6 of [R2]. [Problem Set 4] due. [Notes][pset4solutions]
Feb 23 - Isomorphism theorems, prime ideals, maximal ideals. Sections 7.2-7.4 of [DF], Sections 7-10 of [R2]. [Notes]
Feb 25 - Existence of maximal ideals, monoid rings, polynomial rings. Sections 7.2-7.5 of [DF], Sections 11-21 of [R2]. [Notes]
Feb 27 - [Zoom link] Rings of fractions, review. Sections 7.2-7.5 of [DF] and Sections 11-21 of [R2]. [Problem Set 5] due. [Notes][pset5solutions]
Mar 2 - Midterm Exam 1[Solutions]
Mar 4 - Euclidean domains and principal ideal domains [Notes]
Mar 6 - Examples of non-PID’s. Irreducible and prime elements. [Problem Set 6] due [Notes][pset6solutions]
Mar 9 - PID’s are UFDs. Factorization in Z[i]. Section 8.3 of [DF] and Sections 31-33 of [R2] [Notes]
Mar 11 - Which polynomial rings are UFD’s? Section 9.3 of [DF] and Sections 34-37 of [R2] [Notes]
Mar 13 - Unique factorization in R[x], irreducibility criteria. Sections 9.3-9.4 of [DF] and sections 38-41 of [R2]. [Problem Set 7] due. [Notes][pset7solutions]
***Spring Break***
Mar 23 - Finishing up polynomials. Noetherian rings and the Hilbert Basis theorem. Modules over a ring. Sections 9.4, 9.5, and 10.1 of [DF] and sections 41-48 of [R2]. [Notes]
Mar 25 - Module basics Sections 10.2 and 10.3 of [DF] and Sections 48-52 of [R2]. [Notes]
Mar 27 - Free modules, torsion modules, R-linear independence. Section 10.3 of [DF] and sections 52, 56-57 of [R2]. [Problem Set 8] due. [Notes][pset8solutions]
Mar 30 - Structure theorem for finitely generated modules over PID; Section 12.1 of [DF] and sections 58-61 of [R2]. [Notes]
April 1 - Application to the rational canonical form. Section 12.2 of [DF] and Section 68 of [R2]. [Notes]
April 3 - The Jordan canonical form. Section 12.3 of [DF] and Section 68 of [R2] [Notes]
April 6 - Computations with JCF and RCF [Problem Set 9] due. [Notes][pset9solutions]
April 8 - Second midterm [Solutions]
April 10 - Fields and field extensions. Section 13.1 of [DF] and 1 of [R3] [Notes]
April 13 - [Zoom link]Algebraic and transcendental elements. Sections 13.1-2 of [DF] and 5-9 of [R3]. [Notes]
April 15 - [Zoom link] Finite extensions and compositums of fields. Sections 13.4-5 of [DF] and 12-15 of [R3]. [Notes]
April 17 - Splitting fields and separable polynomials. Sections 13.5-13.6 of [DF] and 23,25,28,29 of [R3], [Problem Set 10] due. [pset10solutions][Notes]
April 20 - Finite fields and cyclotomic fields. Sections 13.5-13.6 of [DF] and 23,25,28,29 of [R3] [Notes]
April 22 - Introduction to Galois theory. Section 14.1-14.2 of [DF]
April 24 - Galois groups of splitting fields. Section 14.2 of [DF] [Problem Set 11] due.
…
May 14 - Final Exam: 8-11am, 136 Davenport Hall.