Is every finite field a quotient ring of ${Z}[x]$? Is every finite field  a quotient ring of ${Z}[x]$? For example, how a field with 27 elements can be written as a quotient ring of ${Z}[x]$?

Every finite field $F$ (of characteristic $p$) is isomorphic to a suitable quotient $F=\Bbb{F}_p[X]/(f)$ where $f \in \Bbb{F}_p[X]$ is an irreducible polynomial. Let $\tilde{f} \in \Bbb{Z}[X]$ be any lift of $f$.
Then $F=\Bbb{Z}[X]/(p, \tilde{f})$.
