Galois theorem in general algebraic extensions

[coquelicot]

I have proved for myself the following theorem, generalizing Galois theorem to general algebraic extensions. My question is: is it true, and is there some reference to this theorem in the literature?

Theorem: Recall that a subfield $M$ of a field $L$ is a perfect closure in $L$ if there is no purely inseparable extension of $M$ inside $L$. In other words, $\text{char}(M) = 0$ or $\text{char}(M) = p > 0$ and all the $p$-roots of elements of $M$ contained in $L$ already belong to $M$.

Assume that $L/K$ is a normal extension of fields. Suppose this extension finite for the sake of simplicity (otherwise, consider only closed groups of automorphisms for the Krull topology). Galois theorem becomes:

The application $M\mapsto H = {\rm Aut}(L/M)$ define a $1\!-\!1$ correspondence, reversing the inclusion, between the perfect closures $M$ in $L$ between $K$ and $L$, and the subgroups $H$ of $\text{Aut}(L/K)$. The invert is given as usual by $H\mapsto M = {\rm Fix}(H)$.

 

[mathwonk]

if i read it correctly, yes this is true and well known probably as long as the subject itself. a reference for it, (the first book i opened from my shelf), is the harvard notes by Richard Brauer, on Galois Theory, from 1957, revised 1963, page 73, in paragraph 9 titled "The main theorem of Galois theory". presumably other references exist easier to find copies of.

 

[mathwonk]

well i could not immediately find another such reference. It seems i was led by your question to pull out the one book on my shelf which treats the theorem in exactly this way. congratulations to you for noticing this generalization of most treatments.

It may be that the results proved about purely inseparable extensions, e.g. in Dummit and Foote implies this version. I.e. given such a general finite normal extension E/F, not necessarily separable, take the fixed field L of the galois group. Then L/F is purely inseparacble, E/L is separable (and normal), and the usual galois theory applied to E/L may yield your version. I.e. perhaps one can show that an intermediare field between E and F is a "perfect closure", iff it contains L?

It also seems that one can apply the usual theory by showing that an intermediate field K, between E and F, is a perfect closure iff E/K is (normal and) separable. This implies that every perfect closure is the fixed filed of some subgroup of the galois group. Conversely the approach of artin to the usual galois theory, shows that E is separable over the fixed field of a subgroup, hence such a fixed field is a perfect closure.

anyway, i think you are quite right, and i think it adds something to look at it in this generality. thank you.

 

[coquelicot]

Yes this is the way I prove the theorem: in an algebraic normal extension $L/K$, a sub extension $F$ is perfect in $L$ if and only if it contains the perfect closure $K_p$ of $K$ in L. Hence everything is usual Galois theory in $L/K_p$ since this extension is separable.

 

[mathwonk]

your question made me think more about non separable extensions and wonder what one knows about the intermediate fields in that case, since it seems the galois theory gives you no handle on them. in particular i wondered if there were any reason for there to be only finitely many intermediate fields, when a google search yielded a theorem of E. Artin that I had forgotten. Namely having only finitely many intermediate fields is equivalent to "simple" extension, i.e. an extension with one generator. so indeed, one may have infinitely intermediate fields in a non separable extension. maybe this is why people often exclude this case in galois theory. as your proof shows, the galois theory only describes the structure of L/Kp, so in some sense one can separate off the study of Kp/K.

 

[coquelicot]

Before I forget it, thank you for your answers mathwonk. Yes, I also think this is the reason for which the Galois correspondence is in general taught only in separable extensions: the study of inseparable extension amounts finally to the study of the separable extension $L/K_p$. Nevertheless, this point is hardly found in the literature.

 

[mathwonk]

yes at harvard we were often spoiled in this regard, since our professors simply wrote up complete treatises of material so that one never had to look in the literature. the notes i cite were professor brauer's notes for his course and were made available for a few dollars at the department. professor mackey wrote out his lectures on complex variables also and they were published in that form, as an essentially perect book on the topic. professor john tate presented an original treatment of infinite dimensional traces in linear algebra (the "tate trace" for "finite - potent" maps), in order to prove the riemann - roch theorem in his course on curves, and subsequently published one of the student's notes from the lecture as a research paper in a prestigious journal. david mumford's notes from his algebraic geometry lectures are still in print as the famous "red book" of algebraic geometry. of course many courses assigned also usual texts, but often these were not at all used, just recommended as extra reading and alternate approaches. the famous book on advanced calculus by loomis and sternberg was originally handed out as notes for the course math 55 there .i still consider the brauer notes as my favorite source on galois theory.