[Math] Proof subextension of Abelian extension is also abelian

Question

This is a property listed on MathWorld:

One nice property of an Abelian extension $K$ of a field $F$ is that
any intermediate subfield $E$, with $F \subset E \subset K$, must be a
Galois extension field of $F$ and, by the fundamental theorem of
Galois theory, also an Abelian extension

What specifically about the fundamental theorem of Galois Theory shows that? To my knowledge, it only shows a one-to-one correspondence from the Galois subgroups and the intermediate fields. How does that make it abelian?

Thanks!

Answer 1

The fundamental theorem tells you that $E/F$ is Galois and that $\text{Gal}(E/F)$ is a quotient of $\text{Gal}(K/F)$--and quotients of abelian groups are abelian.