Math 500, Abstract Algebra I

Fall 2023

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%). You can view all of your scores in the online gradebook.

Weekly homework: These will typically be due on Friday. They are to be turned in on paper at the start of the class period. If you are unable to attend due to illness or quarantine, you can email me a PDF single file with a scan of your HW; if using your phone/tablet, please use an app designed for this purpose, such as Abobe Scan (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: September 27 and Nov 1 (weeks 6 and 11, respectively).

Final exam: The final exam will be Thursday, December 14 at 8am.

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.

Aug 21
Introduction. Sections 1.1-1.5 of [DF], Sections 1-4 of [R1]. Notes.
Aug 23
Isomorphism theorems. Section 3.3 of [DF], Sections 5-10 of [R1]. Notes.
Aug 25
Free groups. Section 6.3 of [DF], Sections 11-14 of [R1]. Notes. HW 1 due. Solutions.
Aug 28
Group presentations; intro to group actions. Sections 6.3 and 1.7 of [DF], Sections 15-17 of [R1]. Notes.
Aug 30
More on group actions. Section 4.1 of [DF], Sections 18-20 of [R1]. Notes.
Sep 1
Applications of group actions, especially conjugacy classes. Sections 4.2-3 of [DF], Sections 21-25 of [R1]. Notes. HW 2 due. Solutions.
Sep 4
Labor Day. No class.
Sep 6
Automorphisms groups of groups; Sections 4.4-5 of [DF], Sections 26-29 of [R1]. Notes.
Sep 8
The Sylow Theorems; Section 4.5 of [DF], Sections 29-34 of [R1]. Notes. HW 3 due. Solutions.
Sep 11
Finitely generated groups and the ascending chain condition; Section 36-38 of [R1]. Notes.
Sep 13
Torsion in abelian groups and direct products; Section 5.1 of [DF] and Sections 38-41 of [R1]. Notes.
Sep 16
Classification of finitely generated abelian groups; Section 5.2 of [DF] and Sections 43-45 of [R1]. Notes. HW 4 due. Solutions.
Sep 18
Composition series and the Jordan-Holder theorem; Sections 5.5 and 3.4 of [DF] and Sections 46-48 of [R1]. Notes.
Sep 20
Solvable and nilpotent groups; Section 6.1 of [DF] and 49-51 of [R1]. Notes.
Sep 22
Review of rings; Sections 7.1 and 7.3 of [DF] and 1-6 of [R2]. Notes. HW 5 due. Solutions.
Sep 25
Isomorphism theorems for rings, more examples; Sections 7.2 and 7.3 of [DF] and 7-10 of [R2]. Notes.
Sep 27
Midterm the First. Handout, exam, and solutions.
Sep 29
Polynomial rings and kinds of ideals; Sections 7.2 and 7.4 of [DF] and 11-16 of [R2]. Notes.
Oct 2
Existence of maximal ideals, rings of fractions; Sections 7.4 and 7.5 of [DF] and Sections 17-21 of [R2]. Notes.
Oct 4
Eucidean Domains amd Principal Ideal Domains; Sections 8.1-8.2 of [DF] and 22-26 of [R2]. Notes.
Oct 6
Examples of non-PIDs; irreducible and prime elements; Sections 8.2-8.3 of [DF] and Sections 27-30 of [R2]. Notes. HW 6 due. Solutions.
Oct 9
PIDs are UFDs, factorization in Z[i]; Section 8.3 of [DF] and Sections 31-33 of [R2]. Notes.
Oct 11
Which polynomial rings are UFDs?; Section 9.3 of [DF] and Sections 34-37 of [R2]. Notes.
Oct 13
Unique factorization in R[x]; irreducibility criteria; Sections 9.3-9.4 of [DF] and Sections 38-41 of [R2]. Notes. HW 7 due. Solutions.
Oct 16
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.
Oct 18
Module basics; Sections 10.2 and 10.3 of [DF] and Sections 48-52 of [R2]. Notes.
Oct 20
Free modules, torsion modules, R-linear independence; Section 10.3 of [DF] and Sections 52, 56-57 of [R2]. Notes. HW 8 due. Solutions.
Oct 23
Classification of modules over a PID; Section 12.1 of [DF] and Sections 58-61 of [R2]. Notes.
Oct 25
Applications of the classification: canonical forms for linear maps. Section 12.2 of [DF] and Section 68 of [R2]. Notes.
Oct 27
More applications to linear maps. Section 12.3 of [DF] and Section 68 of [R2]. Notes. HW 9 due. Solutions.
Oct 30
Fields and field extensions. Section 13.1 of [DF] and 1 of [R3]. Notes.
Nov 1
Midterm the Second. Handout. Exam. Solutions.
Nov 3
Algebraic and transcendental elements. Sections 13.1-13.2 of [DF] and 5-7 of [R3]. Notes.
Nov 6
Finite extensions; compositums of subfields. Section 13.2 of [DF] and 6-9 of [R3]. Notes.
Here are some proofs of the Fundamental Theorem of Algebra.
Nov 8
Splitting fields and separable polynomials. Sections 13.4-13.5 of [DF] and 12-15 of [R3]. Notes.
Nov 10
Finite fields and cyclotomic fields. Sections 13.5-13.6 of [DF] and 23, 25, 28, 29 of [R3]. Notes. HW 10 due. Solutions.
Nov 13
Introduction to Galois Theory. Section 14.1-14.2 of [DF]. Notes.
Nov 15
Galois groups of splitting fields. Section 14.2 of [DF]. Notes.
Nov 17
Foundations of Galois theory. Sections 14.2-3 of [DF]. Notes. HW 11 due. Solutions.
Nov 18
Thanksgiving Break starts.
Nov 26
Thanksgiving Break ends.
Nov 27
Fundamental Theorem of Galois Theory I. Section 14.2 of [DF]. Notes.
Nov 29
Fundamental Theorem of Galois Theory II. Section 14.2 of [DF]. Notes.
Dec 1
Galois groups of polynomials. Section 14.6 of [DF]. Notes. HW 12 due. Solutions.
Dec 4
Solving equations by radicals. Section 14.7 of [DF]. Notes.
Dec 6
The end. Section 14.7 of [DF]. Notes.
Dec 14
Final exam. 8am to 11am. Handout. Solutions to "HW13".