(i) Define a primitive polynomial in $\mathbb{Z}[x]$, and prove that the product of two primitive polynomials is primitive. Deduce that $\mathbb{Z}[x]$ is a unique factorization domain.

(ii) Prove that

$$\mathbb{Q}[x] /\left(x^{5}-4 x+2\right)$$

is a field. Show, on the other hand, that

$$\mathbb{Z}[x] /\left(x^{5}-4 x+2\right)$$

is an integral domain, but is not a field.