\documentclass[11pt, oneside]{article} % use "amsart" instead of "article" for AMSLaTeX format
\usepackage{amsmath,amssymb,amsthm,fullpage}
\usepackage{array}
\newcommand{\bN}{\mathbb{N}}
\newcommand{\bP}{\mathbb{P}}
\newcommand{\bQ}{\mathbb{Q}}
\newcommand{\bR}{\mathbb{R}}
\newcommand{\bZ}{\mathbb{Z}}
\newcommand{\ndiv}{\hspace{-4pt}\not|\hspace{2pt}}
\renewcommand{\theenumi}{\arabic{enumi}}
\renewcommand\labelenumi{\theenumi.}
%%%%%%%%%%
%% Title and author
%%%%%%%%%%
\title{Take Home Final: Due May 3rd 2018\\
MTH 310 001 W18}
\author{YOUR NAME HERE}
\date{}
%% end of preamble
%% beginning of document
\begin{document}
\maketitle
\begin{enumerate}
%%%%%%%
% Question 1
%%%%%%%
\item{} Define a function $\psi: \mathbb{Z}\to \mathbb{Z}$ such that $\psi(n)$ is the number of units in the ring $\mathbb{Z}/n\mathbb{Z}$.
\begin{enumerate}
\item If $p$ is prime, then compute $\psi(p)$.
\item Compute $\psi(6)$, $\psi(10)$, and $\psi(15)$. Conjecture the value of $\psi(pq)$ when $p$ and $q$ are prime and $p\ne q$.
\item A theorem of Euler states that if $gcd(a,n)=1$, then $a^{\psi(n)}\equiv 1 \pmod{n}$. Using this theorem and part (a), prove that every element of $\mathbb{Z}/p\mathbb{Z}$ is a root of the polynomial $x^p-x$ in $\mathbb{Z}/p\mathbb{Z}[x]$. In other words, prove that for any integer $a$ the congruence $a^p\equiv a\pmod{p}$ holds.
\end{enumerate}
\begin{proof}
Type your answer here!
\end{proof}
%%%%%%%
% Question 2
%%%%%%%
\item Let $R$ be a ring with identity and let $n$ be a positive integer. Given an element $a\in R$ we can define $n\cdot a$ to be $a+a+\ldots +a$ with $n$ terms in the sum. We say that $R$ has characteristic $n$ if $n\cdot 1_R=0_R$. If no such $n>0$ exists then $R$ is said to be of characteristic zero.
\begin{enumerate}
\item Prove that the rational numbers $\mathbb{Q}$ are characteristic zero.
\item Let $R$ be a ring with identity with characteristic $n$ with $n>0$.
\begin{enumerate}
\item Prove that $na=0_R$ for every $a\in R$
\item If $R$ is an integral domain, prove that $n$ is prime.
\end{enumerate}
\end{enumerate}
\begin{proof}
Type your answer here!
\end{proof}
%%%%%%%
% Question 3
%%%%%%%
\item Let $a$ be a fixed element of a field $F$ and define a map $ev_a:F[x]\to F$ by $ev_a(f(x))=f(a)$.
\begin{enumerate}
\item Prove that $ev_a$ is a surjective ring homomorphism. We call $ev_a$ an evaluation homomorphism.
\item For an element $a\in F$, let $K_a=\{ g(x) \in F[x] \mid ev_a(g(x))=0_F\}$. Prove that $K_a$ is an ideal in $F[x]$.
\item Describe what it means for a polynomial $h(x)$ to be in $K_a$ in terms of roots of polynomials. Show that the ideal $K_a$ is a principal ideal. What is it generated by?
\end{enumerate}
\begin{proof}
Type your answer here!
\end{proof}
%%%%%%%
% Question 4
%%%%%%%
\item Construct an example of a field with $9$ elements. [Hint: Consider a field extension of $\mathbb{Z}/3\mathbb{Z}$ that contains a root of an irreducible polynomial of degree $2$.]
\begin{proof}
Type your answer here!
\end{proof}
%%%%%%%
% Question 5
%%%%%%%
\item Prove that every ideal in $\mathbb{Z}$ is a principal ideal. [Hint: If $I$ is a nonzero ideal, show that $I$ must contain positive elements and, hence, must contain a smallest positive element $C$ (Why?). Since $c\in I$, every multiple of $C$ is also in $I$; hence, $(c)\subset I$. To show that $I\subset (c)$, let $a$ be an element of $I$. Then $a+cq+r$ with $0\le r