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{\large \textbf{ Math 31: Topics in Algebra \\ 
                   Summer 2019 - Problem Set 7 }}
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{\hfill Due:   Wednesday, August 14}}
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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. 

\vspace{1 cm}

\noindent \textbf{\#1}  How many polynomials of degree exactly $n$ are there over $\mathbb{Z}_p$, where $p$ is prime and $n$ is a non-negative integer? \\

\vspace{1cm}

\noindent \textbf{\#2}  (Polynomial Division)

\begin{enumerate}[a)] 

\item Find the quotient $q(X)$ and remainder $r(X)$ when $a(X) = X^3 + X^2 + X + 1$ is divided by $b(X) = X^2 + 3X + 2$ in $\mathbb{Q}[X]$ and $\mathbb{Z}_5[X]$.  Verify your answers by confirming that $a(X) = b(X)q(X) + r(X)$.

\item Find the quotient $q(X)$ and remainder $r(X)$ when $a(X) = X^3+2$ is divided by $b(X) = 2X^2 + 3X + 4$ in $\mathbb{Q}[X]$, in $\mathbb{Z}_3[X]$, and $\mathbb{Z}_5[X]$.  Verify your answers by confirming that $a(X) = b(X)q(X) + r(X)$.

\end{enumerate} 

\vspace{1cm}

\noindent \textbf{\#3}  (Polynomials over a Ring with Zero Divisors)

\begin{enumerate}[a)]

\item Find all ways to factor $X^2$ into a product of two degree one polynomials over $\mathbb{Z}_9$ and over $\mathbb{Z}_7$.  Why are these answers different?

\item Show that in $\mathbb{Z}_8[X]$, there are infinitely many square roots of 1.

\end{enumerate}

\vspace{1cm}

\noindent \textbf{\#4} (Polynomials over a Field) \\
Let $F$ be a field and let $J$ be an ideal of $F[X]$.  Recall that $F[X]$ is a principal ideal domain: all ideals are principal. 

\begin{enumerate}[a)]

\item Prove that any two generators of $J$ are associates, and so $J$ has a unique monic generator.

\item Prove that $J$ is a prime ideal if and only if it has an irreducible generator.

\end{enumerate}

\vspace{1 cm}

\noindent \textbf{\#5} 
\begin{enumerate}[a)]

\item Factor the following polynomials into irreducibles over $\mathbb{Z}_5:$
\begin{itemize}
\item $X^3 + X^2 + X + 1$
\item $X^5 + 1$
\item $X^4 + 1$
\item $X^4 + 4$
\end{itemize}

\item Show that the following polynomials are irreducible over $\mathbb{Q}:$
\begin{itemize}
\item $3X^4 - 8X^3 + 6X^2 - 4X + 6$ 
\item $\frac{2}{3}X^5 + \frac{1}{2}X^4 - 2X^2 + \frac{1}{2}$
\item $\frac{1}{5}X^4 - \frac{1}{3}X^3 - \frac{2}{3}X + 1$
\end{itemize}

\end{enumerate}

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