For each of the following assertions, provide either a proof or a counterexample as appropriate:

(i) The ring $\mathbb{Z}_{2}[X] /\left\langle X^{2}+X+1\right\rangle$ is a field.

(ii) The ring $\mathbb{Z}_{3}[X] /\left\langle X^{2}+X+1\right\rangle$ is a field.

(iii) If $F$ is a finite field, the ring $F[X]$ contains irreducible polynomials of arbitrarily large degree.

(iv) If $R$ is the ring $C[0,1]$ of continuous real-valued functions on the interval $[0,1]$, and the non-zero elements $f, g \in R$ satisfy $f \mid g$ and $g \mid f$, then there is some unit $u \in R$ with $f=u \cdot g$.