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\centerline{\bf FINAL EXAM--MATH 506A}
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\noindent 1. Classify as completely as you can the fields that are homomorphic images of $\Bbb Q[x,y,z]$, the polynomial ring in three variables over $\Bbb Q$.
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\noindent 2. How many similarity classes are there of $4\times 4$ matrices over $\Bbb Q$ whose minimal polynomial is $(x-1)^2$ and whose characteristic polynomial is $(x-1)^4$?  Give a representative from each class.
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\noindent  3. Let $p$ be a prime and let $G$ be a group of order $p^3$ whose center $Z$ has order $p$.  Find the degrees (=dimensions) of all the irreducible representations of $G$, noting that $G/Z$ is abelian.
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\noindent 4. Give a careful definition of the cohomology groups $H^n(G,A)$ of a finite group $G$ with coefficients in a $G$-module $A$.
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\noindent 5. Let $k$ be an algebraically closed field.  Show that there is no injective $k$-algebra homomorphism from the polynomial ring $k[x_1,\ldots,x_n]$ to $k[x_1,\ldots,x_m]$ if $n>m$.
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\noindent 6. Show that the induced map Spec~$\Bbb Z[i]\rightarrow$ Spec~$\Bbb Z$ arising from the inclusion $\Bbb Z\subset\Bbb Z[i]$ is surjective.
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\noindent 7. Classify the finitely generated projective modules over $\Bbb Z$.
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\noindent 8. Show that the polynomial ring $\Bbb Z[x]$ is not a Dedekind domain but $\Bbb Q[x]$ is.
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