Galois group

The Galois group $\mathrm{Gal}(K/F)$ of a field extension $K/F$ is the group of all field automorphisms $\sigma :K\to K$ 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 ${\sigma }_1,{\sigma }_2\in \mathrm{Gal}(K/F)$, given by ${\sigma }_1:K\to K$ and ${\sigma }_2:K\to K$, the product ${\sigma }_1\cdot {\sigma }_2\in \mathrm{Gal}(K/F)$ is the composite of the two maps ${\sigma }_1\circ {\sigma }_2:K\to K$.

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 $F$.