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\parbox{2.25in}{\bf Math 4720/5210}
\hfill {\Large \bf Homework \#8} \hfill
\parbox{2.25in}{\bf \hfill Due: Wed., Nov. $3^\text{rd}$, 2021}

\vspace{0.2in}

\begin{description}
\item[\#1 Irreducible Problems] Let $\mathbb{F}$ be a field.
   \begin{enumerate}[(a)]
   \item Let $f(x)= a_nx^n+\cdots+a_1x+a_0 \in \mathbb{F}[x]$ be irreducible (where $a_n \not=0$ and $n>0$).\\ \force \hfill Prove that $g(x) = a_0x^n+\cdots+a_{n-1}x+a_n$ (reverse the order of the coefficients) is irreducible in $\mathbb{F}[x]$ as well.\\
    {\it A Hint and a note:} $x^nf(1/x)=g(x)$ is called the {\it reciprocal polynomial} of $f(x)$.
   \item Use part (a) and Eisenstein's criterion to show $h(x)=12x^5-24x^4+6x^2+18x-1$ is irreducible in $\mathbb{Q}[x]$.
   \item Show $\ell(x)=5x^3+9x^2-2x+2$ is irreducible in $\mathbb{Q}[x]$ by reducing modulo $p$ for some prime $p$.
   \item We could also show that $\ell(x)$ is irreducible using the rational root theorem. Sketch out such a proof.\\ 
             {}[You don't actually have go through the trouble of plugging numbers into $\ell(x)$.]
   \end{enumerate}

\item[\#2 Bob the Builder] Construct the field of order 8, $\mathbb{F}_8$. In particular, write down its addition and multiplication tables. Also, we know that $\mathbb{F}_8^\times$ is cyclic. Find a generator.

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{\it Note:} Please adopt notation like that used in our finite fields handout where I construct $\mathbb{F}_4$.

\vspace{0.05in}

\item[\#3 Irreducibly Fun!] First,  a quick note: When looking at factorizations, we can treat associates as essentially equal. So when factoring polynomials in $\mathbb{F}[x]$ (where $\mathbb{F}$ is a field), we can focus on monic polynomials. Why? Given an non-zero polynomial, we can multiply it by the multiplicative inverse of its leading coefficient. This gives us a monic associate (i.e., all non-zero polynomials have a {\it unique} monic associate).

\begin{enumerate}[(a)]
\item Suppose $f(x) \in \mathbb{F}[x]$ and $\deg(f(x)) =4$ or $5$. Explain why if $f(x)$ has no roots and no monic irreducible quadratic factors, then it must be irreducible.
\item List all of the monic irreducible polynomial of degree at most 2 in $\mathbb{Z}_3[x]$.
\item Show that $f(x)=x^5+x^4+2x^3+x^2+x+1$ is irreducible in $\mathbb{Z}_3[x]$.
\item Identify the ring $\mathbb{Z}_3[x]/(x^5+x^4+2x^3+x^2+x+1)$. (i.e., What is it?) {\it Note:} Since we have part (c), no serious computations are needed to answer this question.
\end{enumerate}

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