# Math 448: Modern Algebra I

Final Exam – Thursday, Dec 16, 2021 – 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 PS5, which costs 816 quatloos (hey, do you have any idea how expensive it is to ship a PS5 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 PS5 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 $\forall p\in \mathbb{Z}_5[x], f(p) = [p]$. Prove $f$ is a homomorphism 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 #25.

#### 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