Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $K/F$ be a finite field extension.

If $K/F$ is Galois then it is well known that there is a bijection between subgroups of $Gal(K/F)$ and subfields of $K/F$.

Since finding subgroups of a finite group is always easy (at least in the meaning that we can find every subgroup by brute-force or otherwise) this gives a nice way of finding subfields and proving they are the only ones.

What can we do in the case that $K/F$ is not a Galois extension ? that is: How can I find all subfields of a non-Galois field extension ?

share|improve this question
    
An easy answer to this would be to compute the Galois closure of $K/F$ and consider the subfields of the Galois closure which are in $K$. Would that work for you? –  Patrick Da Silva Oct 14 '12 at 18:54
    
@PatrickDaSilva - what if the extension is not separable ? this answer seems good for the cases of characteristic $0$ and for finite fields so it's very nice on but doesn't answer all of the problem. this is just a question of intrest, I don't need an application for something so any information is good. Thank you for your answer! –  Belgi Oct 14 '12 at 18:58
    
Hm. I must say I assumed the extension was separable because I don't see very often non-separable extensions. =P –  Patrick Da Silva Oct 14 '12 at 19:09
1  
@PatrickDaSilva - quickmeme.com/meme/3rc44q :P –  Belgi Oct 14 '12 at 19:19
    
Very nice meme, I like it –  Patrick Da Silva Oct 15 '12 at 2:40
add comment

2 Answers

up vote 3 down vote accepted

In the inseparable case there is an idea for a substitute Galois correspondence due, I think, to Jacobson: instead of considering subgroups of the Galois group, we consider (restricted) Lie subalgebras of the Lie algebra of derivations. I don't know much about this approach, but "inseparable Galois theory" seems to be a good search term.

share|improve this answer
add comment

An easy answer in the separable case to this would be to compute the Galois closure of K/F and consider the subfields of the Galois closure which are in K.

Hope that helps,

share|improve this answer
add comment

Your Answer

 
discard

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.