Math 448: Modern Algebra I

Final Exam – Thursday, Dec 12, 2019 – 5:15pm

Take Home Final!

Write up and hand in the following. All problems will be graded. Please number them as below, not using the textbook problem number or some other number. Type your solutions in $\LaTeX$ where

  1. On planet Triskelion there are only two kinds of coins, worth 46 quatloos and 26 quatloos respectively. All residents and shopkeepers on the planet have an unlimited number of each type of coin, and only these coins can be used to make purchases.
    1. What is the set of all possible prices an item can have on Triskelion (assuming it can be purchased for its exact price)? For example, you can purchase an item that costs 20 quatloos by giving one 46 quatloo coin to the vender and getting one 26 quatloo coin back as your change, but you can’t purchase an item that costs 1/2 quatloo because there is no way to pay for it exactly. (Prove your answer.)
    2. As Triskelion residents are gamesters, Galt is eager to purchase a new Sony PS4, which costs 816 quatloos (hey, do you have any idea how expensive it is to ship a PS4 to Triskelion?). Can he purchase it, and if so how can Galt pay for it? (i.e., how many coins should he and the vendor exchange of each type)? (Explain your answer.)
    3. If the answer to the previous part is “no”, what is the smallest amount of money he can pay for the PS4 that is more than what it is worth (the vendor doesn’t give discounts). If the answer to the previous part is “yes” what is the least total number of coins that can be exchanged between Galt and the vendor to complete the transaction? (Explain your answer either way.)
  2. Let $R=\mathbb{Z}_5[x]/(3x^5+x^4+3x+1)$.
    1. How many elements are in the ring $R$? (Explain.)
    2. Is $R$ a field? (Prove your answer.)
    3. Find a monic degree 2 irreducible polynomial in $\mathbb{Z}_5[x]$ whose equivalence class is a zero divisor in $R$. (Prove your answer.)
    4. Find a monic degree 2 irreducible polynomial in $\mathbb{Z}_5[x]$ whose equivalence class is a unit in $R$ and find its multiplicative inverse. (Prove your answer.)
    5. Define $f:\mathbb{Z}_5[x]\to R$ by $\forall p\in \mathbb{Z}_5[x], f(p) = [2p]$. Prove $f$ is an isomorphism and $R\cong \mathbb{Z}_5[x]/\text{Ker}(f)$.
  3. Exercise 7.4 #8.
  4. Exercise 7.4 #18.
  5. Exercise 7.4 #19.
  6. Exercise 8.1 #6. Note: $U_{32}$ is what we would call $U\left(\mathbb{Z}_{32}\right)$.
  7. Exercise 8.1 #10.
  8. Exercise 8.1 #24.
  9. Exercise 8.1 #32.
  10. Exercise 7.5 #24.

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
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: $\mathbb{Z}_p$ 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
1 5 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: $F[x]\left/p(x)\right.$ 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
28 Thu, Dec 3 Section 8.1: Congruence and Lagrange’s Theorem
27 Tue, Dec 5 Section 7.5: Symmetric and Alternating Groups & Putnam!
29 Thu, Dec 12 Final Exam