\input amstex
\input amssym.def
\loadmsbm
\magnification=\magstep1
\nopagenumbers
\bigskip
\centerline{\bf SAMPLE FINAL PROBLEMS--MATH 506A}
\vskip .4in
\noindent 1. State a version of Hilbert's Nullstellensatz that holds over an arbitrary basefield $k$.
\vskip .4in
\noindent 2. Give a precise statement of the rational canonical form for square matrices over a field $k$.
\vskip .4in
\noindent 3. Classify the prime ideals in the ring $\Bbb Z[i]$ of Gaussian integers.
\vskip .4in
\noindent 4. Give an example of a Noetherian domain of dimension one that is not a PID.
\vskip .4in
\noindent 5. Compute the cohomology groups $H^n(\Bbb Z_m,A)$ of the cyclic group $G=\Bbb Z_m$ of order $m$ with coefficients in any $G$-module $A$.
\vskip .4in
\noindent 6. Prove or disprove:  two finite groups $G,H$ are isomorphic if and only if their complex group algebras $\Bbb CG,\Bbb CH$ are isomorphic as rings.
\vskip .4in
\noindent 7. The ring $R=\Bbb Z[\sqrt{-7}]$ is not quite a Dedekind domain; how can this ring be enlarged to make it such a domain?
\vskip .4in
\noindent 8. Compute the Krull dimension of the polynomial ring $\Bbb Z[x,y,z]$.
\end