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

If $L/K$ and $F/L$ are Galois extensions inside $\mathbb{C}$, must $F/K$ be a Galois extension?

share|cite|improve this question
No, this is not true. – Reader Oct 29 '12 at 23:01
Could you give a counter-example? – Montez Oct 29 '12 at 23:05
I gave an example for if $L/F$ and $F/K$ are Galois then $L/K$ need not be Galois, but that is not what you are asking? – Reader Oct 29 '12 at 23:35
up vote 6 down vote accepted

Take the extension $$\mathbb{Q}\subset\mathbb{Q}(\sqrt{2}))\subset\mathbb{Q}(\sqrt[4]{2})$$ Then each of the intermediate steps are Galois as they are of degree two, but the total degree 4 extension $\mathbb{Q}\subset\mathbb{Q}(\sqrt[4]{2})$ is not Galois as some of the roots of the minimal polynomial of $\sqrt[4]{2}$ over $\mathbb{Q}$ is not in $\mathbb{Q}(\sqrt[4]{2})$.

share|cite|improve this answer

Consider the extension $\mathbb Q\subset\mathbb Q(\sqrt[4]{2})\subset \mathbb Q(\sqrt[4]{2},i) $. You have that $\mathbb Q(\sqrt[4]{2})/\mathbb Q$ is not Galois since it is not normal. Yu have to enlarge $\mathbb Q(\sqrt[4]{2})$ over $\mathbb Q$ in order to get Galois extension.

share|cite|improve this answer
But the question is: If $L/K$ and $F/L$ are Galois extensions inside $\mathbb{C}$, must $F/K$ be a Galois extension? Rather than whether a Galois extension can live inside a bigger Galois extension. – Montez Oct 29 '12 at 23:11

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.