Math 448: Modern Algebra I

Assignment #7 – Thursday, Sep 19, 2019
  • Read the proof of Bezout’s Lemma (either the proof of Theorem 1.2 in the textbook or the proof in the lecture notes.
  • Write up and hand in your proofs of the following theorems.
    1. Let $a,b\in\mathbb{Z}. If $b\mid a$ and $a\neq 0$ then $b\leq |a|$.
    2. Let $b\in\mathbb{Z}$ and $b\neq 0$. Then $$\gcd(b,0)=\left|b\right|$$
    3. Exercise 1.2 #8
    4. Exercise 1.2 #20

Course Handouts and References

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

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
  • 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

Mathematical Writing and Typesetting

  • Overleaf – a free website where you can easily produce LaTeX math documents through a web browser
  • LaTeX Homework Template – click this link to start a new assignment.
  • Dr. Monks’ LaTeX Style – a sample document illustrating the features of our assignment style
  • LaTeX for Windows
  • MikTeX – install this and TeXnicCenter for LaTeX on your Windows computer
  • TeXnicCenter – install this and MikTeX for LaTeX on your Windows computer
  • LaTeX for Mac
  • MacTeX – Install this for LaTeX on your Mac computer
Tentative Schedule
#DateTopic
1 Tue, Aug 27 Introduction
2 Thu, Aug 29 Logic and Proof
3 Tue, Sep 3 Appendix B: Sets and Functions
4 Thu, Sep 5 Appendix D: Equivalence Relations
5 Tue, Sep 10 Appendix C: Induction
6 Thu, Sep 12 Section 1.1: Division Algorithm
7 Tue, Sep 17 Section 1.2: Divisibility
8 Thu, Sep 19 Section 1.3: Primes and Unique Factorization
9 Tue, Sep 24 Section 2.1-2.2: Congruence and Modular Arithmetic
10 Thu, Sep 26 Section 2.3: Zp when p is prime
11 Tue, Oct 1 Section 3.1: Rings
12 Thu, Oct 3 Section 3.2: Basic Properties of Rings
13 Tue, Oct 8 Section 3.3: Isomorphism and Homomorphism
14 Thu, Oct 10 Section 4.1: Polynomials and Division Algorithm
15 Thu, Oct 17 Section 4.2: Divisibility in F[x]
16 Tue, Oct 22 Section 4.3: Irreducibles and Unique Factorization
17 Thu, Oct 24 Section 4.4: Polynomial Functions, Roots, and Divisibility
18 Tue, Oct 29 Section 5.1: Congruence in F[x]
19 Thu, Oct 31 Section 5.2: Modular Arithmetic in F[x]
20 Tue, Nov 5 Section 5.3: F[x]/p(x) when p(x) is Irreducible
21 Thu, Nov 7 Section 6.1: Ideals and Congruence
22 Tue, Nov 12 Section 6.2: Quotient Rings and Homomorphisms
23 Thu, Nov 14 Section 7.1: Groups
24 Tue, Nov 19 Section 7.2: Basic Properties of Groups
25 Thu, Nov 21 Section 7.3: Subgroups
26 Tue, Nov 26 Section 7.4: Isomorphisms and Homomorphisms
27 Tue, Dec 3 Section 7.5: Symmetric and Alternating Groups
28 Thu, Dec 5 Review & Putnam!
29 Thu, Dec 12 Final Exam