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\large
\centerline{\rm \bf  Math 301   Homework 7 (due on October 18, 2023)}\vskip.1in

\vskip.2in
\noindent{\Large\bf Section 4.3}
\vskip.1in
\noindent{\bf 6.}  Show that $x^2+1$ is irreducible in $\Q  [x]$. \vskip.2in
\vskip.1in
 
\noindent{\bf   12.}    Express $x^4-4$ as a product of irreducibles in $\Q  [x],\ \R[x]$, and in $ \C[x]$. 
\vskip.1in
\noindent{\bf 14.} Show that $x^2+x$ can be factored in two ways in $\Z_6[x]$ as the product of non-constant  
 polynomials that are not units and not associates of $x$ or $x+1$.
\vskip.1in

\noindent{\bf 22.} 


\begin{itemize} 

\item[(a)] Show that $x^3+a$ is reducible in $\Z_3[x]$ for each $a\in \Z_5$.

\item[(b)]   Show that $x^5+a$ is reducible in $\Z_5[x]$ for each $a\in \Z_5$.



\end{itemize} 

\vskip.2in
\noindent{\Large\bf Section 4.4}

\noindent{\bf 2.} Find the remainder when $f(x)$ is divided by $g(x):$
\begin{itemize} 

\item[(a)]  $f(x)=x^{10}+x^8$ and $g(x)=x-1$ in $\Q[x]$.

\item[(d)]  (Typo corrected on Oct 16, 11am) $f(x)=2x^5-3x^4+x^3+\mathbf{2x}+3$ and $g(x)=x-3$ in $\Z_5[x]$.

\end{itemize} 

\vskip.2in
\noindent{\bf 8.} Determine if the given polynomial is irreducible.
\vskip.1in

\begin{itemize} 

 
\item[(c)] $x^2+7 $ in $\C[x]$.  


 
\item[(d)]  $2 x^3 + x^2 + 2 x + 2$ in $\Z_5[x]$.


 
\item[(e)]   $ x^3-9 $ in $\Z_{11}[x]$.

\item[(f)]   $x^4+x^2+1 $  in $\Z_{3}[x]$.

 \end{itemize}
\vskip.1in
\noindent{\bf 10.}  Find a prime $p>5$ such that $x^2+1$ is reducible in $\Z_p[x]$.  

\vskip.1in

\noindent{\bf 16.}  Let $f(x),g(x)\in F[x]$ have degree $\le n$ and let $c_0,\dots,c_n$ be distinct elements of $  F $.  If  $f(c_i)=g(c_i)$ for $i=0,\dots, n$, prove that $f(x)=g(x)$ in $ F [x]$.
\end{document}



