nLab Galois extension

Classically, “Galois extension” refers to a class of extensions of fields.

An extension $K\subset L$ of fields is Galois if it is normal and separable. In that case, the automorphism group $Aut_K(L)$ of $K$-automorphisms of $L$ is called the Galois group and often denoted by $Gal(L:K)$.

There is a famous Galois theory for such extensions.

Now there are numerous generalizations, including Hopf-Galois extensions, coalgebra Galois extensions etc.