Galois group

Given a field extension $Q \subseteq K$ , its Galois group is

G a l (K | Q) = {φ : K \to K, field homom such that φ |_{K} = i d} .

This field homorphisms are called $Q$ -automorphisms.

The Galois group of a polynomial $p (x) \in Q [x]$ is $G a l (K | Q)$ where $K$ is the splitting field of $p (x)$ .

If $p (x) \in Q [x]$ , $α \in K$ and $φ \in G a l (K | Q)$ then

φ (p (α)) = p (φ (α)),

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

G a l (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 $S_{n} \subseteq G a l (K | Q)$ , for example, consider $x^{4} - 5 x^{2} + 6$ . The splitting field is $K = Q (\sqrt{2}, \sqrt{3})$ and its Galois group cannot contain an element $φ$ 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