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\large
\centerline{\rm \bf  Math 301   Homework 8 (due on October 25, 2023)}\vskip.1in

\vskip.2in
\noindent{\Large\bf Section 4.5}
\vskip.1in
\noindent{\bf 4.} Show that each polynomial is irreducible in $\Q  [x]$, as in Example 3.  \vskip.2in
\vskip.1in
\begin{itemize} 

\item[(a)]   $f(x)=x^4+2x^3+x+1$. 

\item[(b)]  $f(x)=x^4-2x^2+8x+1$



\end{itemize} 
 
\noindent{\bf   8.}   Give an example of  a polynomial $f(x)\in \Z[x]$ and a prime $p$  such  that $f(x)$  is  reducible in $\Q  [x]$ but  $\overline{f}(x)$ is irreducible in $\Z_p[x]$. Does this contradict Theorem 4.25? 
\vskip.1in
\noindent{\bf 18.}     Which of these polynomials are  irreducible in $\Q  [x]$?\vskip.1in
 


\begin{itemize} 

\item[(a)]  $f(x)=x^4-x^2+1$. 

\item[(b)]   $f(x)=x^4+x +1$. 



\end{itemize} 


\vskip.2in
\noindent{\bf 19(b).} Write the polynomial $x^7-2x^6-6x^4-15x^2-33x-9$ as a product of    irreducible polynomials  in $\Q  [x]$.


\vskip.2in
\noindent{\Large\bf Section 4.6}

\noindent{\bf 1 (b).}  Find all roots in $\C$ of $f(x)=x^4-2x^3-x^2+6x-6$;  given  $1+i$ is a root.
\vskip.2in
\noindent{\bf 4.} Factor $x^2+x+ 1+i  $ in $\C[x]$. 


\vskip.2in
\noindent{\Large\bf Section 5.1}

\noindent{\bf 4.} Show that, under congruence modulo $x^3+2x+1$ in $\Z_3[x]$, there are exactly 27 distinct congruence classes.
\vskip.1in


\noindent{\bf 8.}  Prove or disprove: If $p(x)$ is relatively prime to $k(x)$  and $f(x)k(x)\equiv g(x)k(x)\ (\mod  p(x))$, then $f(x) \equiv g(x)\  (\mod  p(x))$.  
\vskip.1in

\noindent{\bf 10.}  Prove or disprove: If  $p(x)$ is irreducible in $\F[x]$  and $f(x)g(x)\equiv 0_{\F}\  (\mod  p(x))$, then $f(x) \equiv 0_{\F}\  (\mod  p(x))$ or $g(x) \equiv 0_{\F}\  (\mod  p(x))$. 
\vskip.1in


\noindent{\bf 12.}   	If  $f(x)$ is relatively prime to $p(x)$, prove that there is a polynomial $g(x) \in \F[x]$  such that $f(x)g(x) \equiv  1_{\F}\  (\mod  p(x))$. 

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