I know that $\mathbb{Z}[x]/\langle4, x^2+x+1\rangle$ is isomorphic to $(\mathbb{Z}/4\mathbb{Z})[X]/\langle x^2+x+1\rangle $. Since $(\Bbb Z/4\Bbb Z)[X]$ is not a PID, irreducibility/reducibility of $x^2+x+1$ is not enough to say whether or not this is a field. My idea is to prove that this ring is not a domain. For example, $(\mathbb{Z}/4\mathbb{Z})[X]/\langle x^2+x+1\rangle$ is not a domain if $2$ is not in the ideal $\langle x^2+x+1\rangle$ but I can't prove this.