Galois group

The Galois group $\operatorname{Gal}(K/F)$ of a field extension $K/F$ is the group of all field automorphisms $\sigma\colon K \to K$ of $K$ which fix $F$ (i.e., $\sigma(x) = x$ for all $x \in F$ ). The group operation is given by composition: for two automorphisms $\sigma_1, \sigma_2 \in \operatorname{Gal}(K/F)$ , given by $\sigma_1\colon K \to K$ and $\sigma_2\colon K \to K$ , the product $\sigma_1 \cdot \sigma_2 \in \operatorname{Gal}(K/F)$ is the composite of the two maps $\sigma_1 \circ \sigma_2\colon K \to K$ .

The Galois group of a polynomial $f(x) \in F[x]$ is defined to be the Galois group of the splitting field of $f(x)$ over $F$ .