Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?


share|cite|improve this question
At the moment this has a tag "finite-fields", which isn't really relevant to the question. – KCd Apr 1 '12 at 5:03
up vote 6 down vote accepted

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.

share|cite|improve this answer
How do you know that $E \subset F$ is normal? – Larry X Apr 1 '12 at 4:44
@Larry: the fundamental theorem tells you that $E/F$ is Galois iff $\text{Gal}(K/E)$ is normal in $\text{Gal}(K/F)$, which is automatically true if $\text{Gal}(K/F)$ is abelian. – Qiaochu Yuan Apr 1 '12 at 4:46
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. – Alex Youcis Apr 1 '12 at 4:46

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.