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{\large \textbf{ Math 31: Topics in Algebra \\ 
                   Summer 2019 - Problem Set 6 }}
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{\hfill Due:   Wednesday, August 7}}
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%\textbf{keywords:} \textit{homomorphism theorem, graph automorphisms} \\
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\textit{Instructions:} Write or type your answers neatly, and staple all pages before passing in your assignment.  Use complete sentences where appropriate.  Show your work; little credit will be given for solutions without work or justification. 

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\noindent \textbf{\#1} Let $A$ be a \emph{principal ideal domain}: an integral domain in which every ideal is principal (eg. we proved that $\mathbb{Z}$ is a PID using the division algorithm).  Let $J = \langle x \rangle$ be a prime ideal of $A$.  Show that $J$ is maximal (if $J \subseteq I$ for some ideal $I$, then $I \in \{J, A\}$).

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\noindent \textbf{\#2} 
\begin{enumerate}[a)]

\item Let $A$ be a finite commutative ring with unity.  Prove that every nonzero element is either a unit or a zero divisor.

\item Let $A$ be a finite integral domain.  Show that if there exist some nonzero $x, y \in A$ such that $(x+y)^2 = x^2+y^2$, then $\Char(A) = 2$.

\end{enumerate}

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\noindent \textbf{\#3} 
\begin{enumerate}[a)]

\item Prove that for all nonzero integers $a$ and $b$, $\gcd(a, b) = \gcd(a,b+ax)$ for all $x \in \mathbb{Z}$.

\item Recall that for two ideals $I$ and $J$ in a ring $A$, $I+J = \{x+y \, : \, x \in I, y \in J\}$ is also an ideal of $A$.  Prove that for nonzero integers $a$ and $b$ wth $\gcd(a, b) = d$, $\langle a \rangle + \langle b \rangle = \langle d \rangle$.

\end{enumerate}

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\noindent \textbf{\#4} 
\begin{enumerate}[a)]

\item Prove that if $a^2 \equiv 1 \pmod 2$ then $a^2 \equiv 1 \pmod 4$.

\item Let $p$ and $q$ be distinct primes. Prove $p^{q-1} + q^{p-1} \equiv 1 \pmod{pq}$.

\end{enumerate}

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