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\title{Math 332: Homework 9}
%\author{Your name here --- delete the % at the start of the line}
\date{13.IV.15}

\begin{document}
\maketitle

\begin{exc}
Let $p(x,y,z) = 2x^2-3xy^3z+4y^2z^5$ and $q(x,y,z) = 7x^2+5x^2y^3z^4-3x^2z^3$ be polynomials in $\ZZ[x,y,z]$.  Let $\bar{p}(x,y,z)$ and $\bar{q}(x,y,z)$ denote the images of $p(x)$ and $q(x)$ in $(\ZZ/3\ZZ)[x]$ under the canonical reduction map.
\begin{enumerate}[(a)]
\item Write each of $p$, $q$, $\bar{p}$, $\bar{q}$ as a polynomial in $x$ with coefficients in $\ZZ[y,z]$ or $(\ZZ/3\ZZ)[y,z]$.
\item Find the degree of $p$, $q$, $\bar{p}$, $\bar{q}$.
\item Find the degree of $p$, $q$, $\bar{p}$, $\bar{q}$ in each of the variables $x$, $y$, $z$.
\item Compute $pq$, $\bar{p}\bar{q}$ and find the degree of each in each of the variables $x$, $y$, $z$.
\item Write $pq$, $\bar{p}\bar{q}$ as a polynomial in $z$ with coefficients in $\ZZ[x,y]$ or $(\ZZ/3\ZZ)[x,y]$.
\end{enumerate}
\end{exc}

\begin{exc}
Prove that the ideals $(x)$ and $(x,y)$ are prime in $\QQ[x,y]$, but only $(x,y)$ is maximal.
\end{exc}

\begin{prob}
Prove that a polynomial ring in infinitely many variables
\[
  R[x_1,x_2,x_3,\ldots]
\]
where $R$ is any commutative ring (with $1\ne 0$) contains ideals which are not finitely generated.  (Thus $R[x_1,x_2,x_3,\ldots]$ is an example of a \emph{non-Noetherian ring}.)
\end{prob}

\begin{chal}
An ideal $I$ in $R[x_1,\ldots,x_n]$ is called a \emph{homogeneous ideal} if whenever $p(x)\in I$, then each of the homogeneous components of $p$ is also in $I$.  Prove that an ideal is a homogeneous ideal if and only if it may be generated by homogeneous polynomials.
\end{chal}

\begin{prob}
Let $f(x)$ be a polynomial in $F[x]$, $F$ a field.  Prove that $F[x]/(f(x))$ is a field if and only if $f(x)$ is irreducible.
\end{prob}

\begin{prob}
Briefly describe the ring structure of each of the following rings:
\begin{enumerate}[(a)]
\item $\ZZ[x]/(2)$,
\item $\ZZ[x]/(x)$,
\item $\ZZ[x]/(x^2)$,
\item $\ZZ[x,y]/(x^2,y^2,2)$.
\end{enumerate}
Show that $\alpha^2 = 0$ or $1$ for every $\alpha$ in the last ring and determine those elements with $\alpha^2=0$.
\end{prob}

\begin{prob}
Let $F$ be a field and let $R$ be the set of polynomials in $F[x]$ whose coefficient of $x$ is $0$.  Prove that $R$ is a subring of $F[x]$.  Use the equation $x^6 = (x^2)^3 = (x^3)^2$ to prove that $R$ is not a UFD.
\end{prob}

\begin{exc}
Determine (with proof) whether the following polynomials are irreducible in the rings indicated.
\begin{enumerate}[(a)]
\item $x^2+x+1$ in $(\ZZ/2\ZZ)[x]$
\item $x^3+x+1$ in $(\ZZ/3\ZZ)[x]$
\item $x^4+1$ in $(\ZZ/5\ZZ)[x]$
\item $x^4+10x^2+1$ in $\ZZ[x]$
\end{enumerate}
\end{exc}

\begin{prob}
Show that the polynomial $(x-1)(x-2)\cdots (x-n)+1$ is irreducible in $\ZZ[x]$ for all $n\ge 1$, $n\ne 4$.
\end{prob}

\begin{prob}
Prove that $\RR[x]/(x^2+1)$ is a field isomorphic to $\CC$.
\end{prob}

\begin{prob}
Let $\FF_{11} = \ZZ/11\ZZ$.  Prove that $K_1 = \FF_{11}[x]/(x^2+1)$ and $K_2 = \FF_{11}[y]/(y^2+2y+2)$ are both fields with $121$ elements.  Prove that the map which sends $p(\bar{x})\in K_1$ to $p(\bar{y}+1)\in K_2$ is well-defined and gives a field isomorphism $K_1\to K_2$.
\end{prob}

\end{document}