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\large
\centerline{\rm \bf  Math 301   Homework 9 (due on November 1, 2023)}\vskip.1in

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\noindent{\Large\bf Section 5.2}
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\noindent{\bf 2.}   Write out the addition and multiplication tables for the congruence-class ring $\Z_3[x]/(x^2+1)$.
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\noindent{\bf 6.}   Determine the rules for addition and multiplication of congruence classes in $\Q  [x]/(x^2-2)$. 
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\noindent{\bf 14.}  In each part explain why $[f(x)]$ is a unit in $\F[x]/(p(x))$ and find its inverse. 
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\begin{itemize} 

\item[(a)]   $ [f(x)]=[2x-3]\in \Q  [x]/(x^2-2)$. 

\item[(b)]  $ [f(x)]=[x^2+x+1]\in \Z_3[x]/(x^2+1)$. 



\end{itemize} 
\noindent{\bf  16.}   Show that $\Q  [x]/(x^2-2)$ is a field. 
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\noindent{\Large\bf Section 5.3}
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 \noindent{\bf 1.} Determine whether the given congruence-class ring is a field. Justify your answer. 
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\begin{itemize} 

\item[(a)]     $\Z_3[x]/(x^3+2x^2+x+1)$

\item[(b)]    $\Z_5[x]/(2x^3-4x^2+2x+1)$.



\end{itemize} 
\noindent{\bf 2.}  

 \begin{itemize} 

\item[(a)]     Verify that $\Q  (\sqrt{2}) = \{r + s\sqrt{2} \ r, s \in \Q   \}$ is a subfield of $\R$.


\item[(b)]    Show that $\Q  (\sqrt{2}) $ is isomorphic to $\Q  [x]/(x^2-2) $.

\end{itemize} 
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\noindent{\bf 6.}  Let $p(x)$ be irreducible in $\F[x]$. If $[f(x)]\neq  [0_{\F}]$ in $\F[x]/(p(x))$ and $h(x)\in \F [x]$, prove that there exists $g(x) \in \F [x]$  such that $[f(x)][g(x)] = [h(x)]$ in $\F[x]/(p(x))$.  [{\it Hint :} Theorem 5.10 and Exercise 12 (b) in Section 3.2]
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\noindent{\bf 8.}   If $p(x)$ is an  irreducible quadratic polynomial in $F[x]$, show that $F[x]/(p(x))$ contains all the roots of $p(x)$.
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