# Math 448: Modern Algebra I

Assignment #7 – Thursday, Sep 23, 2021
• Write up and hand in your proofs of the following theorems.
1. Let $b\in\mathbb{Z}$ and $b\neq 0$. Then $\gcd(b,0)=\left|b\right|$
2. Let $n$ be any integer. Then $\gcd(n,n+1)=1$.
3. Let $a,b,k\in\mathbb{Z}$ with $a\neq 0$ or $b\neq 0$. Then $$\gcd(a,b)=\gcd(a,b+ak)$$

#### Course Handouts and References

• – check here often for revised lecture notes for our course
• Course Syllabus
• Proofs – notes and handouts from my Math 299 course on mathematical proofs

### Math 448 Online

• Math 448 Online Classroom
• If we have to pivot to remote instruction this semester, click here to enter our Zoom classroom.
• Zoom help – Zoom for Scranton students, how to install, how to use, documentation.

#### Software and Handouts

• Lurch – a math word processor that can check your proofs! I designed and wrote this software with former Scranton math major Nathan Carter
• 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.
• Toy Proof Talk Slides – slides from a talk I gave on Toy Proofs at the 2009 Joint Mathematics Meetings
• Group Explorer – amazing group visualization and exploration software from Nathan Carter

#### Homework Assignment Template

Tentative Schedule
#DateTopic
1 Tue, Aug 31 Introduction
2 Thu, Sep 2 Logic and Proof
3 Tue, Sep 7 Appendix B: Sets and Functions
4 Thu, Sep 9 Appendix D: Equivalence Relations
5 Tue, Sep 14 Appendix C: Induction
6 Thu, Sep 16 Section 1.1: Division Algorithm
7 Tue, Sep 21 Section 1.2: Divisibility
8 Thu, Sep 23 Section 1.3: Primes and Unique Factorization
9-10 Tue, Sep 28 Section 2.1-2.2: Congruence and Modular Arithmetic
11 Thu, Sep 30 Section 2.3: $\mathbb{Z}_p$ when $p$ is prime
12 Tue, Oct 5 Section 3.1: Rings
13 Thu, Oct 7 Section 3.2: Basic Properties of Rings
14 Thu, Oct 14 Section 3.3: Isomorphism and Homomorphism
15 Tue, Oct 19 Section 4.1: Polynomials and Division Algorithm
16 Thu, Oct 21 Section 4.2: Divisibility in $F[x]$
17 Tue, Oct 26 Section 4.3: Irreducibles and Unique Factorization
18 Thu, Oct 28 Section 4.4: Polynomial Functions, Roots, and Divisibility
19 Tue, Nov 2 Section 5.1: Congruence in $F[x]$
20 Thu, Nov 4 Section 5.2: Modular Arithmetic in $F[x]$
21 Tue, Nov 9 Section 5.3: $F[x]\left/p(x)\right.$ when $p(x)$ is Irreducible
22 Thu, Nov 11 Section 6.1: Ideals and Congruence
23 Tue, Nov 16 Section 6.2: Quotient Rings and Homomorphisms
24 Thu, Nov 18 Section 7.1: Groups
25 Tue, Nov 23 Section 7.2: Basic Properties of Groups
26 Tue, Nov 30 Section 7.3: Subgroups
27 Thu, Dec 2 Section 7.4: Isomorphisms and Homomorphisms & Putnam!
28 Tue, Dec 7 Section 8.1: Congruence and Lagrange’s Theorem
29 Thu, Dec 9 Section 7.5: Symmetric and Alternating Groups