Let $\displaystyle J $ be the set of all polynomials with zero constant term in $\displaystyle \mathbb{Z}[x] $.

(a) Show that $\displaystyle J $ is the principal ideal $\displaystyle (x) $ in $\displaystyle \mathbb{Z}[x] $.

(b) Show that $\displaystyle \mathbb{Z}[x]/J $ consists of an infinite number of distinct cosets.

So for (a) suppose it is not? For (b) you would also use contradiction?