(a) Prove that every principal ideal domain is a unique factorization domain.

(b) Consider the ring $R=\{f(X) \in \mathbb{Q}[X] \mid f(0) \in \mathbb{Z}\}$.

(i) What are the units in $R$ ?

(ii) Let $f(X) \in R$ be irreducible. Prove that either $f(X)=\pm p$, for $p \in \mathbb{Z}$ a prime, or $\operatorname{deg}(f) \geqslant 1$ and $f(0)=\pm 1$.

(iii) Prove that $f(X)=X$ is not expressible as a product of irreducibles.