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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!

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    At the moment this has a tag "finite-fields", which isn't really relevant to the question.2012-04-01

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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.

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    Because you know that $E/F$ will be normal if and only if the subgroup of $\text{Gal}(K/F)$ corresponding to $E$ is normal. But, $\text{Gal}(K/F)$ is abelian so every subgroup is normal.2012-04-01