I'm working on this problem : In $Z[x]$, find an ideal $P$ such that $R/P$ consists of $4$ elements.
My idea is to try and construct a suitable quotient of $Z[x]$ and an appropriate ideal generated by an irreducible polynomial, so that $Z[x]/P$ is isomorphic to the field with $4$ elements $F_4$. However, I'm only familiar with the theory that $Z_p[x]/P$, where $P$ is the ideal generated by an irreducible polynomial of degree $n$, is isomorphic to the field with $p^n$ elements.
How can I carry this over to a similar type of quotient of $Z[x]$ by a single ideal? I thought to maybe take the quotient $Z[x]/4Z$ first to achieve $Z_4%$, then quotient this new ring by an ideal generated by a linear polynomial (degree $1$) to construct a field with $4^1 = 4$ elements. However, this is sort of a "double quotient" that makes it difficult for me to see what the original ideal $P$ that we quotient $Z[x]$ by would be.