Let $f = X^3 + 3X + 2 $ in $\mathbb{Z}[X]$
1) Show that the factor ring $\mathbb{Q}[X]/\langle f \rangle$ is a field
2) Show that in the factor ring $\mathbb{Z}[X]/\langle f \rangle$ the element $X$ is nonzero and not invertible and deduce that $\mathbb{Z}[X]/\langle f \rangle$ is not a field.
How do I prove those two statements?
For the second one I was thinking that I should use a proof by contradiction, but I don't completely grasp how to do it.