## 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.

- Let $p\in\mathbb{N}$. If $1<p$ and $2^p-1$ is prime then $p$ is prime.
- Let $n\in\mathbb{Z}$. Then $n$ is odd if and only if $n\underset{4}{\equiv}1$ or $n\underset{4}{\equiv}-1$.
- 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*.) - 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 $0_R$.
- 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.)