\input amstex
\input amssym.def
\nopagenumbers
\magnification=\magstep1
\centerline{\bf HW \#7}
\bigskip
\centerline {\bf Math 506A}
\bigskip
\noindent 1. (20 points) Let $R$ be the ring of continuous real-valued functions on the unit interval $S=[0,1]$.  Use the compactness of $S$ to show that given any proper ideal $I$ of $R$, the set $Z(I)\subseteq S$ of common zeros of functions in $I$ is nonempty.  On the other hand, show that distinct radical ideals $I$ of $R$ can have the same $Z(I)$.
\vskip .3in
\noindent 2. Let $k$ be a field.  Show that the ideal $(x^n,x^{n-1}y,\ldots,y^n)$ in $k[x,y]$ cannot be generated by fewer than $n+1$ elements.  
\vskip .3in
\noindent 3. (20 points) Classify the primes in the ring $R=\Bbb Z[i]=\{a+bi: a,b\in\Bbb Z\}$, as follows.  You may assume that $R$ is a PID and thus a UFD.  Recall that the {\sl norm} $N(a+bi)$ of a Gaussian integer $a+bi$ is the product $(a+bi)(a-bi)=a^2+b^2$ and that $N(zw) = N(z)N(w)$ for $z,w\in R$.  Using norms, show that every prime in $R$ is either a multiple of a prime $p\in\Bbb Z$ by a unit ($\pm1$ or $\pm i$) or else has prime norm.  Show that a prime $p\equiv3 \mod 4$ in $\Bbb Z$ remains prime in $R$.  Show that a prime $p\equiv1 \mod 4$ in $\Bbb Z$ factors in $R$ as $(a+bi)(a-bi)$ for some $a,b$, by first showing that the equation $x^2+1=0$ has a solution in the modular integers $\Bbb Z_p$.  Finally, deal with the case $p=2$.
\end