Construction of a finite field

Question

Let $Z[X]$ denote the ring of polynomials in $X$ with integer coefficients .Find an ideal $I$ in $Z[X]$ such that $Z[X]/I$ is a field of order $4$.

My attempt:I know that if $F$ is a field & $f(x)$ is irreducible in $F[X]$ then $F[X]/\langle f(x)\rangle$ is a field.If we take $F =Z_p$ & $f(x)$ as an $2$ degree polynomial then such a construction is possible.But not sure how to proceed here

Answer 1

$\Bbb Z[X]/\langle p \rangle \cong \Bbb Z_p[X]$. So what happens when $I$ contains both $p$ and a monic degree two irreducible polynomial?