Answer the following questions, fully justifying your answer in each case.

(i) Give an example of a ring in which some non-zero prime ideal is not maximal.

(ii) Prove that $\mathbb{Z}[x]$ is not a principal ideal domain.

(iii) Does there exist a field $K$ such that the polynomial $f(x)=1+x+x^{3}+x^{4}$ is irreducible in $K[x]$ ?

(iv) Is the ring $\mathbb{Q}[x] /\left(x^{3}-1\right)$ an integral domain?

(v) Determine all ring homomorphisms $\phi: \mathbb{Q}[x] /\left(x^{3}-1\right) \rightarrow \mathbb{C}$.