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   \Blatt{1}
   \Abgabe{12/11/2001}


   \Nr
   Let $K$ be any field, $I\unlhd K[x_1,\ldots,x_n]=K[\underline{x}]$ an ideal. \\
   Proof the following statements, or find counterexamples.
   \begin{enumerate}
   \item If $V(I)=\A^n(K)$, then $I=(0)$.
   \item If $V(I)=\emptyset$, then $I=K[\underline{x}]$.
   \item If $I$ is a prime ideal, then $V(I)$ is
     irreducible.\footnote{A non-empty topological space $V$ is called
       \emph{irreducible} if it is not the union of two proper subsets
       which are both closed.}
   \end{enumerate}
   Is it possible to weaken respectively to strengthen the hypotheses such
   that the results remain respectively become true?

   \Nr
   Let $K$ be algebraically closed, and $I\unlhd K[\underline{x}]$
   be a radical ideal. \\
   Show that $I=\bigcap\big\{\m \big| I\subseteq
     \m\in\mSpec(K[\underline{x}])\big\}$, where
   $\mSpec(K[\underline{x}])$ is the set of maximal ideals of $K[\underline{x}]$.
   Interprete this geometrically. 
   
   \Nr
   Let $K$ be an infinite field, $V=V\big(y-x^2\big)\subset\A_K^2$ and
   $V'=V\big(xy-1\big)\subset\A_K^2$. Show
   that $K[V]:=K[x,y]/I(V)\cong K[t]$ and
   $K\big[V'\big]:=K[x,y]/I\big(V'\big)\cong K\big[t,\tfrac{1}{t}\big]$.

   \Nr
   With the aid of the ray tracer \texttt{surf} and the library
   \texttt{surf.lib} \textsc{Singular} can draw curves in $A_\R^2$ and
   surfaces in $A_\R^3$. Once the library is loaded and 
   the ring and the polynomial are defined, then the command \texttt{plot}
   may be used to draw  the zero-set of the polynomial, e.~g.~
   \begin{center}
     \begin{minipage}{8cm}
       \tt
       LIB "\hspace*{0.01mm}surf.lib";\\
       ring r=0,(x,y,z),lp;\\
       poly f=x2-yz;\\
       plot(f);
     \end{minipage}
   \end{center}

   Draw the curve given by $f=y^2-x^2-x^3$ and
   the surface given by
   $g=(x+y+z-1)\cdot(x-y-z-1)\cdot(y-x-z-1)\cdot(z-x-y-1)\cdot(x+y+z+1)\cdot(x-y-z+1)\cdot(y-x-z+1)\cdot(z-x-y+1)+\big(x^2+y^2+z^2-1\big)\cdot\big(x^2+y^2+z^2-1\big)\cdot\big(x^2+y^2+z^2-2\big)\cdot\big(x^2+y^2+z^2-2\big)$.\footnote{For $g$ use the command  \texttt{plot(g,"\hspace*{0.01mm}scale\_x=0.2;scale\_y=0.2;");}.}  
   

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