Math 500, Abstract Algebra I

Spring 2025

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. This corresponds roughly to Chapters 1-8 and 10-14 of the textbook.

Prerequisites: Undergraduate linear and abstract algebra (basics of groups, rings, fields, vectors spaces, etc.), for example as covered in Math 416 and Math 417.

Required text: Dummit and Foote, Abstract Algebra, 3rd Edition, 944 pages, Wiley 2003. The Grainger Engineering Library has a copy on reserve for in-library use.

Suplemental 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 Policies

Overall grading: Your course grade will be based on homework (40%), two in-class midterm exams (15% each), and a comprehensive final exam (30%).

Weekly homework: These will always be due on Friday, at the start of the class period. Submission will be by upload to Gradescope, which you can access from the Canvas page for this course. Gradescope will accept any PDF file and will ask you to indicate where your answer to each problem starts for ease of grading. If you are scanning handwritten homework with your phone/tablet, please use an app designed for this purpose, such as Abobe Scan (iOS, Android) or the Gradescope app (iOS, Android). Late homework will not be accepted; however, your lowest two homework grades will be dropped, so you are effectively allowed two infinitely late assignments. Collaboration on homework is permitted, nay encouraged! However, you must write up your solutions individually and understand them completely.

In-class midterms: These two 50 minute exams will be held in our usual classroom on the following Wednesdays: February 26 and April 9 (weeks 6 and 11, respectively).

Final exam: The final exam will be Thursday, May 15 at 1:30pm.

Missed exams: There will typically be no make-up exams. Rather, in the event of a valid illness, accident, or family crisis, you can be excused from an exam so that it does not count toward your overall average. I reserve final judgment as to whether an exam will be excused. All such requests should be made in advance if possible, but in any event no more than one week after the exam date.

Cheating: Cheating is taken very seriously as it takes unfair advantage of the other students in the class. Penalties for cheating on exams, in particular, are very high, typically resulting in a 0 on the exam or an F in the class.

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.

Detailed Schedule

Includes scans of my lecture notes and the homework assignments. Here [DF] refers to the text by Dummit and Foote, and [R1], [R2], and [R3] refers to the three parts of Rezk's notes.

Jan 22
Introduction. Sections 1.1-1.5 of [DF], Sections 1-4 of [R1]. Notes.
Jan 24
Isomorphism theorems. Section 3.3 of [DF], Sections 5-10 of [R1]. Notes.
Jan 27
Free groups. Section 6.3 of [DF], Sections 11-14 of [R1]. Notes.
Jan 29
Group presentations; intro to group actions. Sections 6.3 and 1.7 of [DF], Sections 15-17 of [R1]. Notes.
Jan 31
More on group actions. Section 4.1 of [DF], Sections 18-20 of [R1]. Notes. HW 1 due. Solutions.
Feb 3
Applications of group actions, especially conjugacy classes. Sections 4.2-3 of [DF], Sections 21-25 of [R1]. Notes.
Feb 5
Automorphisms groups of groups; Sections 4.4-5 of [DF], Sections 26-29 of [R1]. Notes.
Feb 7
The Sylow Theorems; Section 4.5 of [DF], Sections 29-34 of [R1]. Notes. HW 2 due. Solutions.
Feb 10
Finitely generated groups and the ascending chain condition; Section 36-38 of [R1]. Notes.
Feb 12
Torsion in abelian groups and direct products; Section 5.1 of [DF] and Sections 38-41 of [R1]. Notes.
Feb 14
Classification of finitely generated abelian groups; Section 5.2 of [DF] and Sections 43-45 of [R1]. Notes. HW 3 due. (as TeX).
Feb 17
Composition series and the Jordan-Holder theorem; Sections 5.5 and 3.4 of [DF] and Sections 46-48 of [R1]. Draft notes.
Feb 19
Solvable and nilpotent groups; Section 6.1 of [DF] and 49-51 of [R1]. Draft notes.
Feb 21
Review of rings; Sections 7.1 and 7.3 of [DF] and 1-6 of [R2]. Draft notes. HW 4 due.
Feb 24
Isomorphism theorems for rings, more examples; Sections 7.2 and 7.3 of [DF] and 7-10 of [R2]. Draft notes.
Feb 26
Midterm the First. Handout, practice exam, and solutions.
Feb 28
Polynomial rings and kinds of ideals; Sections 7.2 and 7.4 of [DF] and 11-16 of [R2]. Draft notes.
Mar 3
Existence of maximal ideals, rings of fractions; Sections 7.4 and 7.5 of [DF] and Sections 17-21 of [R2]. Draft notes.
Mar 5
Eucidean Domains amd Principal Ideal Domains; Sections 8.1-8.2 of [DF] and 22-26 of [R2]. Draft notes.
Mar 7
Examples of non-PIDs; irreducible and prime elements; Sections 8.2-8.3 of [DF] and Sections 27-30 of [R2]. Draft notes. HW 5 due.
Mar 10
PIDs are UFDs, factorization in Z[i]; Section 8.3 of [DF] and Sections 31-33 of [R2]. Draft notes.
Mar 12
Which polynomial rings are UFDs?; Section 9.3 of [DF] and Sections 34-37 of [R2]. Draft notes.
Mar 14
Unique factorization in R[x]; irreducibility criteria; Sections 9.3-9.4 of [DF] and Sections 38-41 of [R2]. Draft notes. HW 6 due.
Mar 17
Spring Break starts.
Mar 21
Spring Break ends.
Mar 24
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]. Draft notes.
Mar 26
Module basics; Sections 10.2 and 10.3 of [DF] and Sections 48-52 of [R2]. Draft notes.
Mar 28
Free modules, torsion modules, R-linear independence; Section 10.3 of [DF] and Sections 52, 56-57 of [R2]. Draft notes. HW 7 due.
Mar 31
Classification of modules over a PID; Section 12.1 of [DF] and Sections 58-61 of [R2]. Draft notes.
Apr 2
Applications of the classification: canonical forms for linear maps. Section 12.2 of [DF] and Section 68 of [R2]. Draft notes.
Apr 4
More applications to linear maps. Section 12.3 of [DF] and Section 68 of [R2]. Draft notes. HW 8 due.
Apr 7
Fields and field extensions. Section 13.1 of [DF] and 1 of [R3]. Draft notes.
Apr 9
Midterm the Second.
Apr 11
Algebraic and transcendental elements. Sections 13.1-13.2 of [DF] and 5-7 of [R3]. Draft notes.
Apr 14
Finite extensions; compositums of subfields. Section 13.2 of [DF] and 6-9 of [R3]. Draft notes.
Here are some proofs of the Fundamental Theorem of Algebra.
Apr 16
Splitting fields and separable polynomials. Sections 13.4-13.5 of [DF] and 12-15 of [R3]. Draft notes.
Apr 18
Finite fields and cyclotomic fields. Sections 13.5-13.6 of [DF] and 23, 25, 28, 29 of [R3]. Draft notes. HW 9 due.
Apr 21
Introduction to Galois Theory. Section 14.1-14.2 of [DF]. Draft notes.
Apr 23
Galois groups of splitting fields. Section 14.2 of [DF]. Draft notes.
Apr 25
Foundations of Galois theory. Sections 14.2-3 of [DF]. Draft notes. HW 10 due.
Apr 28
Fundamental Theorem of Galois Theory I. Section 14.2 of [DF]. Draft notes.
Apr 30
Fundamental Theorem of Galois Theory II. Section 14.2 of [DF]. Draft notes.
May 2
Galois groups of polynomials. Section 14.6 of [DF]. Draft notes. HW 11 due.
May 5
Solving equations by radicals. Section 14.7 of [DF]. Draft notes.
May 7
The end. Section 14.7 of [DF]. Draft notes.
May 15
Final exam. 1:30pm to 4:30am.