- Prove the theorems in this Lurch document. Save it in your Dropbox folder in a subfolder named
assignment 04(lower case, one space, a zero before the 3, no # or hyphens). No AI or Google! Yuk! 🤮 - Putnam Training! (optional bonus points) Look at the problems in Round 1 of this year’s USAMTS competition. We will follow the same rules and regulations of the competition itself, but you will only submit any solutions you find to me, not to the contest (since you aren’t eligible!). Solutions should be handed in a folder named
USAMTS Round 1in your Dropbox folder before 10pm Eastern on October 13, 2026. You don’t need to verify them in Lurch, but you should write up your solutions either in Lurch or LaTeX. Game on!
Math 448: Modern Algebra I
Assignment #4 – Due Tuesday, Sep 15, 2026
Course Handouts and References
- Lecture Notes – check here often for revised lecture notes for our course
- Course Syllabus
- Proofs – notes and Lurch contexts from my Math 299 course on mathematical proofs
Software and Handouts
- Lurch – a math word processor that can check your proofs!
- Overleaf – a free website where you can easily produce LaTeX math documents through a web browser
- Toy Proofs – a “toy” proof system I developed to introduce students to the concept of formal proofs.
- Group Explorer – amazing group visualization and exploration software from Nathan Carter
Homework Assignment Template
- LaTeX Homework Template – click this link to start a new assignment.
- Documentation for our LaTeX Homework Template – a sample document illustrating the features of our assignment style
- Overleaf LaTeX Documentation – a nice general help file and reference guide to using LaTeX.
- MikTeX – install this and TeXnicCenter for LaTeX on your Windows computer
- TeXnicCenter – install this and MikTeX for LaTeX on your Windows computer
- MacTeX – Install this for LaTeX on your Mac computer
LaTeX for Windows
LaTeX for Mac
| # | Date | Topic |
|---|---|---|
| 1 | Tue, Sep 1 | Introduction; Logic and Proof |
| 2 | Thu, Sep 3 | Appendix B,C,D (equivalence relations & induction) |
| 3 | Tue, Sep 4 | Section 1.1: Division Algorithm |
| 4 | Thu, Sep 10 | Section 1.2: Divisibility |
| 5 | Tue, Sep 15 | Section 1.3: Primes and Unique Factorization |
| 6 | Thu, Sep 17 | Section 2.1-2.2: Congruence and Modular Arithmetic |
| 7 | Tue, Sep 22 | Section 2.3: $\mathbb{Z}_p$ when $p$ is prime |
| 8 | Thu, Sep 24 | Section 3.1: Rings |
| 9-10 | Tue, Sep 29 | Section 3.2: Basic Properties of Rings |
| 11 | Thu, Oct 1 | Section 3.3: Isomorphism and Homomorphism |
| 12 | Tue, Oct 6 | Section 4.1: Polynomials and Division Algorithm |
| 13 | Thu, Oct 8 | Section 4.2: Divisibility in $F[x]$ |
| 14 | Thu, Oct 15 | Section 4.3: Irreducibles and Unique Factorization |
| 15 | Tue, Oct 20 | Section 4.4: Polynomial Functions, Roots, and Divisibility |
| 16 | Thu, Oct 22 | Section 5.1: Congruence in $F[x]$ |
| 17 | Tue, Oct 27 | Section 5.2: Modular Arithmetic in $F[x]$ |
| 18 | Thu, Oct 29 | Section 5.3: $F[x]\left/p(x)\right.$ when $p(x)$ is Irreducible |
| 19 | Tue, Nov 3 | Section 6.1: Ideals and Congruence |
| 20 | Thu, Nov 5 | Section 6.2: Quotient Rings and Homomorphisms |
| 21 | Tue, Nov 10 | Section 7.1: Groups |
| 22 | Thu, Nov 12 | Section 7.2: Basic Properties of Groups |
| 23 | Tue, Nov 17 | Section 7.3: Subgroups |
| 24 | Thu, Nov 19 | Section 7.4: Isomorphisms and Homomorphisms |
| 25 | Tue, Nov 24 | Section 8.1: Congruence and Lagrange’s Theorem |
| 26 | Tue, Dec 1 | Section 7.5: Symmetric and Alternating Groups |
| 27 | Thu, Dec 3 | Putnam!|
| 28 | Tue, Dec 8 | |
| 29 | Thu, Dec 10 |