(a) Let $f(x) \in \mathbb{F}_{q}[x]$ be a polynomial of degree $n$, and let $L$ be its splitting field.

(i) Suppose that $f$ is irreducible. Compute $\operatorname{Gal}(f)$, carefully stating any theorems you use.

(ii) Now suppose that $f(x)$ factors as $f=h_{1} \cdots h_{r}$ in $\mathbb{F}_{q}[x]$, with each $h_{i}$ irreducible, and $h_{i} \neq h_{j}$ if $i \neq j$. Compute $\operatorname{Gal}(f)$, carefully stating any theorems you use.

(iii) Explain why $L / \mathbb{F}_{q}$ is a cyclotomic extension. Define the corresponding homomorphism $\operatorname{Gal}\left(L / \mathbb{F}_{q}\right) \hookrightarrow(\mathbb{Z} / m \mathbb{Z})^{*}$ for this extension (for a suitable integer $m$ ), and compute its image.

(b) Compute $\operatorname{Gal}(f)$ for the polynomial $f=x^{4}+8 x+12 \in \mathbb{Q}[x]$. [You may assume that $f$ is irreducible and that its discriminant is $576^{2}$.]