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\begin{document}

\begin{center}
\textbf{Fall 2019, Math 620:\ Week 9 Problem Set} \\
\textbf{Due:\ Thursday, November 7th, 2019} \\
\textbf{Classifying Finite Fields}
\end{center}


\bigskip
\noindent
\textbf{Homework problems.}
You must submit \emph{all} homework problems in order to receive full credit.  

\begin{enumerate}[(H1)]
\item 
Factor $f(x) = x^5 + x^4 + 1$ over $\FF_2$, $\FF_4$, and $\FF_8$.  

\begin{proof}

\end{proof}


\item 
Determine how many elements of $\FF_{32}$ are primitive.  Hint:\ no excessive calculations needed!  

\begin{proof}

\end{proof}


\item 
Find a formula for the product of all nonzero elements of $\FF_q$.  

\begin{proof}

\end{proof}


\item 

\begin{enumerate}
\item 
Let $a(n)$ denote the number of degree-$n$ irreducible polynomials over $\FF_2$.  Prove that
$$2^n = \sum_{d \mid n} d\! \cdot\! a(d).$$
Hint: use the theorem about how $x^{2^d} - x$ factors over $\FF_2$.  

\begin{proof}

\end{proof}


\item 
Find the number of irreducible polynomials over $\FF_2$ with degree exactly 31.  

\begin{proof}

\end{proof}


\item 
Find the number of irreducible polynomials over $\FF_2$ with degree exactly 21.  

\begin{proof}

\end{proof}

\end{enumerate}


\item 
Determine whether each of the following statements is true or false.  Prove your assertions.  

\begin{enumerate}[(a)]
\item 
No finite field is algebraically closed (recall that a field $F$ is \emph{algebraically closed} if every polynomial in $F[x]$ has a root in $F$).  

\begin{proof}

\end{proof}


\item 
The finite field $\FF_{p^r}$ has a subfield (that is, a subring that is also a field) with exactly~$q$ elements if and only if $q = p^t$ for some $t \le r$.  

\begin{proof}

\end{proof}

\end{enumerate}

\end{enumerate}


\bigskip
\noindent
\textbf{Challenge problems.}
Challenge problems are not required for submission, but bonus points will be awarded for submitting a partial attempt or a complete solution.  

\begin{enumerate}[(C1)]
\item 
Fix a finite field $\FF_q$, and let $a(n)$ denote the number of irreducible polynomials over $\FF_q$ of degree exactly $n$.  Prove that 
$$\lim_{n \to \infty} \frac{a(n)}{q^n} = 0,$$
meaning that irreducible polynomials are ``sparse'' in $\FF_q[x]$.  

\begin{proof}

\end{proof}

\end{enumerate}

\end{document}

