 Galois closure of a field extension

The Galois closure of a finite separable extension $L/K$ is the smallest Galois extension in a fixed separable closure containing $L$. It is generated by all the conjugates of $L$. If $G=\mathrm{Gal}(M/K)$ and $H=\mathrm{Gal}(M/L)$, their subgroup core is trivial. This is distinct from the existing group-theoretic normal closure.