Let $\mathbb{F}_{q}$ be a finite field with $q$ elements and $\overline{\mathbb{F}}_{q}$ its algebraic closure.

(i) Give a non-zero polynomial $P(X)$ in $\mathbb{F}_{q}\left[X_{1}, \ldots, X_{n}\right]$ such that

$$P\left(\alpha_{1}, \ldots, \alpha_{n}\right)=0 \quad \text { for all } \alpha_{1}, \ldots, \alpha_{n} \in \mathbb{F}_{q}$$

(ii) Show that every irreducible polynomial $P(X)$ of degree $n>0$ in $\mathbb{F}_{q}[X]$ can be factored in $\overline{\mathbb{F}}_{q}[X]$ as $(X-\alpha)\left(X-\alpha^{q}\right)\left(X-\alpha^{q^{2}}\right) \cdots\left(X-\alpha^{q^{n-1}}\right)$ for some $\alpha \in \overline{\mathbb{F}}_{q}$. What is the splitting field and the Galois group of $P$ over $\mathbb{F}_{q}$ ?

(iii) Let $n$ be a positive integer and $\Phi_{n}(X)$ be the $n$-th cyclotomic polynomial. Recall that if $K$ is a field of characteristic prime to $n$, then the set of all roots of $\Phi_{n}$ in $K$ is precisely the set of all primitive $n$-th roots of unity in $K$. Using this fact, prove that if $p$ is a prime number not dividing $n$, then $p$ divides $\Phi_{n}(x)$ in $\mathbb{Z}$ for some $x \in \mathbb{Z}$ if and only if $p=a n+1$ for some integer $a$. Write down $\Phi_{n}$ explicitly for three different values of $n$ larger than 2 , and give an example of $x$ and $p$ as above for each $n$.