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\centerline{\bf Sample Final Problems--Math 506}
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\noindent 1. State the version of the Nullstellensatz that holds for arbitrary basefields.
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\noindent 2. Let $f:A\rightarrow B$ be a homomorphism of rings.  Show that the induced map $f^*:\,\,$Spec$(B)\rightarrow\,\,$Spec$(A)$ is continuous with respect to the Zariski topology on both spectra.
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\noindent 3. State the theorem on existence of primary decompositions of ideals in a Noetherian ring, indicating also the two features of this decomposition that are uniquely determined by the ideal.
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\noindent 4. Give an example of a Noetherian domain of dimension one that is not a PID.
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\noindent 5. Give a sufficient condition for a polynomial $q\in\Bbb Z[x]$ to have a solution in $\Bbb Z_p$, the ring of $p$-adic integers, for some prime $p$.
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\noindent 6. Show that $\Bbb Z[i]$, the ring of Gaussian integers, is a faithfully flat extension of $\Bbb Z$, first by showing that it is flat, and then verifying its faithfulness by showing that the induced map on prime spectra is surjective.
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\noindent 7. Give a necessary and sufficient condition in terms of ideal factorization for one nonzero ideal $I$ in a Dedekind domain $A$ to divide another.
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\noindent 8. Show that there is no surjective morphism from $K^m$ to $K^n$ for any algebraically closed field $K$ if $m<n$.
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