Given a field extension $Q\subseteq K$, its Galois group is
$$ Gal(K|Q)=\{\varphi:K\rightarrow K \text{, field homom such that } \varphi|_K=id\}. $$This field homorphisms are called $Q$-automorphisms.
The Galois group of a polynomial $p(x)\in Q[x]$ is $Gal(K|Q)$ where $K$ is the splitting field of $p(x)$.
If $p(x)\in Q[x]$, $\alpha \in K$ and $\varphi \in Gal(K|Q)$ then
$$ \varphi(p(\alpha))=p(\varphi(\alpha)), $$so given a $Q$-automorphism of the splitting field of a polynomial $p(x)$, it sends roots of $p(x)$ in roots of $p(x)$. Therefore
$$ Gal(K|Q)\subseteq S_n, $$where $n$ is the degree of the polynomial and $S_n$ is the group of permutations (symmetric group). But not necessarily $S_n \subseteq Gal(K|Q)$, for example, consider $x^4-5x^2+6$. The splitting field is $K=Q(\sqrt{2},\sqrt{3})$ and its Galois group cannot contain an element $\varphi$ sending $\sqrt{2}$ to $\sqrt{3}$.
Therefore, we can understand the Galois group as an action on the space of the roots of the polynomial.
The Galois group of a polynomial is just $S_n$ if the polynomial is irreducible. See this video
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Author of the notes: Antonio J. Pan-Collantes
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