# For each normal extension of a field, whose Galois group is commutative, each intermediate extension is also normal.

Let $K$ be a normal extension of the field $F$, and let the Galois group $G(K,F)$ be an Abelian group. Prove that each intermediate extension $E$ is also a normal extension.

EDIT: All fields here are of characteristic $0$, otherwise we would need to require the extension to be separable.

Every sub-group of an Abelian is normal. By the fundamental theorem of Galois theory $G(E,F)\triangleleft G(K,F)\Leftrightarrow K^{G(E,F)}=E\triangleleft K$. q.e.d.