Math 448: Modern Algebra I

Midterm Take-Home – Thursday, Oct 17, 2019

Take-Home Midterm Exam

All of the following problems will be graded. No collaboration allowed. You can still ask me questions as usual. Be sure to include the problem statement above your proofs.

Part A

Write up and hand in a proof of each of the following. (+1 point each) All proofs should be in semi-formal proof style typed in $\LaTeX$, and the pdf handed in in Dropbox. These solutions are due on Thursday, October 17, by 6:03pm.

  1. Let $p\in\mathbb{N}$. If $1<p$ and $2^p-1$ is prime then $p$ is prime.
  2. Let $n\in\mathbb{Z}$. Then $n$ is odd if and only if $n\underset{4}{\equiv}1$ or $n\underset{4}{\equiv}-1$.
  3. Let $(R,+,\cdot)$ be a ring and define $S=\{r\in R : r= \vphantom{r}^-\!r\}$.  Then $S$ is a subring of $R$. (Don’t be sloppy with mixing subtraction and additive inverse in this one.)
  4. Let $F$ be a field, $R$ a ring, and $f:F\to R$ a ring homomorphism.  If $f$ maps two distinct elements of $F$ to $0_R$ then $f$ maps every element of $F$ to $0_R$.
  5. Let $R$ be a ring and $\phi:R[x]\to R$ be the function that maps each polynomial to its constant term. Then $\phi$ is a surjective ring homomorphism.

Part B

Finish solving the Mango Sharing Bonus Problem: (up to +5 points) Write up and hand in your proof to Problem #4/1/31 on the current round of the USAMTS. Your proof should be in traditional informal word-wrapped proof style like what you would do on the Putnam, and typed in $\LaTeX$. Your solution will be graded on a scale of 0-5 by Putnam scoring standards. Your solution pdf must be placed in Dropbox before 3PM local time on Tuesday, October 15. (It can be in a separate file from the solutions to Part A since they aren’t due until Thursday.)

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
27 Tue, Dec 3 Section 7.5: Symmetric and Alternating Groups
28 Thu, Dec 5 Review & Putnam!
29 Thu, Dec 12 Final Exam