$3 . \mathrm{II} . 11 \mathrm{C}$

(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.

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