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\begin{flushright}
  Your Name Here
\end{flushright}




\begin{center}
  \textbf{Math 71}\\ Homework 8
\end{center}



\begin{enumerate}

\item [pXXX, \#1] Let $R$ be an integral domain and $I \subset R$ an
  ideal.  Let $f\mapsto \overline f$ denote the reduction homomorphism
  $R[x] \to (R/I)[x]$. For $f \in R[x]$, show that
  $R[x]/(I,f) \cong (R/I)[x]/(\overline f)$.  Hint: You should verify
  that there is a natural surjective homomorphism
  $R[x] \to (R/I)[x] \to (R/I)[x]/(\overline f)$.  Give a careful
  argument to show that the map has the desired kernel.

  A standard example is take $f\in \Z[x]$.  Then
  $\Z[x]/(n, f(x)) \cong \Z/n\Z[x]/(\overline f)$ which gives you
  information about the ideal $(n,f)$ in $\Z[x]$ in terms of
  $\Z/n\Z[x]/(\overline f)$. You will no doubt find other uses.
  
\item [p298. \#5.] Prove that $(x,y)$ and $(x,y,2)$ are prime ideals
  in $\Z[x,y]$, but only the latter is maximal.

\item [p298, \#13.]  Show that the rings $F[x,y]/(y^2-x)$ and
  $F[x,y]/(y^2-x^2)$ are not isomorphic for any field $F$.

\item [p301, \#3] Let $F$ be a field and $x$ and indeterminate,
  $f \in F[x]$.  Show that $F[x]/(f)$ is a field if and only if $f$ is
  irreducible in $F[x]$. (Use Proposition 7 in \S8.2)


\item [p301, \#5] If $p(x) \in F[x]$, exhibit all the ideals in
  $F[x]/(p(x))$ using the factorization of $p$ in $F[x]$.

\item [p301, \#7] Determine all the ideals of the ring
  $\Z[x]/(2,x^3+1)$.
  


  
\end{enumerate}

 

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