- Prove the theorems and answer the questions in this Lurch document. Save it in your Dropbox folder in a subfolder named
assignment 08(lower case, one space, a zero before the 8, no # or hyphens). No AI or Google! Yuk! 🤮 - Putnam Training! (optional bonus points) Continue to optionally work on the bonus problems in Round 1 of this year’s USAMTS competition. Game on!
Math 448: Modern Algebra I
Assignment #8 – Due Thursday, Oct 1, 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 |