Galois Extension and Galois Group

A finite field extension $K/k$ is separable if every element of $K$ has a separable minimal polynomial over $k$.

It is normal if every $k$-embedding $K\hookrightarrow \bar k$ has image $K$.

It is Galois if it is both normal and separable; then the Galois group is $\mathrm{Gal}(K/k):=\mathrm{Aut}_k(K).$

In the letter: $\Gamma=\mathrm{Gal}(K/k)$ acts on split data and is used in descent/twisting and in the $L$-group .