(i) Give an example of a field $F$, contained in $\mathbb{C}$, such that $X^{4}+1$ is a product of two irreducible quadratic polynomials in $F[X]$. Justify your answer.

(ii) Let $F$ be any extension of degree 3 over $\mathbb{Q}$. Prove that the polynomial $X^{4}+1$ is irreducible over $F$.

(iii) Give an example of a prime number $p$ such that $X^{4}+1$ is a product of two irreducible quadratic polynomials in $\mathbb{F}_{p}[X]$. Justify your answer.

[Standard facts on fields, extensions, and finite fields may be quoted without proof, as long as they are stated clearly.]