The question I'm trying to answer reads:
Describe the embeddings Gal$(f) \subset S_2$ and Gal$(f) \subset S_9$ for $f = X^2 - X - 1$ and $f = X^9-X ∈ _3[X]$
but I don't know how to interpret it. Does it mean Gal$(X^2 -X-1) \subset S_2$ and Gal$(X^9-X) \subset S_9$ or also, in addition, Gal$(X^2 -X-1) \subset S_9$ and Gal$(X^9 -X) \subset S_2$? I'm completely new to Galois theory, and might lack some relevant knowledge about finite fields..
As Jyrki points out in the comments, if $f \in k[X]$ is a nonconstant separable polynomial with (distinct) roots $\alpha_1, \dots, \alpha_n \in \overline{k}$, the extension $k(\alpha_1,\dots, \alpha_n)/k$ is Galois and a morphism
$$ \sigma : k(\alpha_1,\dots, \alpha_n)/k \to k(\alpha_1,\dots, \alpha_n) /k $$
is determined by $\sigma(\alpha_i)$ for each $i$. Note that field morphisms are injective so $\sigma(\alpha_i) \neq \sigma(\alpha_j)$ when $i\neq j$. This defines a permutation on $\{1,\dots, n\}$ via
$$ \tau_\sigma(i) = j \iff \sigma(\alpha_i) = \alpha_j $$
and the function $\sigma \in Gal(f) \mapsto \tau_\sigma \in S_n$ is a group morphism. Moreover, it is injective: if $\tau_\sigma = 1$, then $\sigma(\alpha_i) = \alpha_i$ for all $i$ and thus $\sigma = id$.
Hence $Gal(f) \hookrightarrow S_{\deg f}$ for any $f \in k[X]$ (separable and non-constant) via the aforementioned assignment.