Complete splitting is preserved by Galois closure
= Complete splitting is preserved by Galois closure
If $M/K$ is the <Galois closure> of a finite extension $L/K$, a prime splits completely in $L$ exactly when it splits completely in $M$. Complete splitting means every right action of $D$ on $H\backslash G$ fixes each coset, so $D$ lies in the <subgroup core> of $H$. That core is trivial because the conjugates of $L$ generate $M$.