Even in our day of sophisticated search engines, it still seems that the success of a search often turns on knowing exactly the right keyword.
I just followed up on Sylvain Bonnot's comment above. The property of a field extension $K/F$ that for all subextensions $L$ we have $K^{\operatorname{Aut}(K/L)} = L$ is apparently most commonly called Dedekind. This terminology appears in Exercise V.9 of Bourbaki's Algebra II, where the reader is asked to show that if $L/K$ is a nonalgebraic Dedekind extension and $T$ is a transcendence basis, then $L/K(T)$ must have infinite degree. Ironically, this is exactly what I could show in my note. One can (in the general case, even...) immediately reduce to the case $T = \{t\}$ and then the exercise is saying that the function field $K(C)$ of an algebraic curve (again, it is no loss of generality to assume the function field is regular by enlarging $K$) is not Dedekind over $K$. This is kind of a strange coincidence! [However, the proof I give is openly geometric so is probably not the one that N.B. had in mind...]
It also appears in
MR0067098 (16,669f)
Barbilian, D.
Solution exhaustive du problème de Steinitz. (Romanian. Russian, French summary)
Acad. Repub. Pop. Române. Stud. Cerc. Mat. 2, (1951). 195–259 (misprinted 189–253).
In this paper, the author shows that $L/K$ is a Dedekind extension iff for all subextensions $M$, the algebraic closure $M^*$ of $M$ in $L$ is such that $M^*/M$ is Galois in the usual sense: i.e., normal and separable. (This is a nice fact, I suppose, and I didn't know it before, but it seems that the author regarded this as a solution of the problem of which extensions are Dedekind. I don't agree with that, since it doesn't answer my question!)
Apparently one is not supposed to read the above paper but rather this one:
MR0056588 (15,97b)
Krull, Wolfgang
Über eine Verallgemeinerung des Normalkörperbegriffs. (German)
J. Reine Angew. Math. 191, (1953). 54–63.
Here is the MathSciNet review by E.R. Kolchin (who knew something about transcendental
Galois extensions!):
The author reviews a definition and some results of D. Barbilian [Solutia exhaustiva a problemai lui Steinitz, Acad. Repub. Pop. Române. Stud. Cerc. Mat. 2, 189--253 (1950), unavailable in this country], providing proofs which are said to be simpler, and further results. Let L be an extension of a field K. Then L is called normal over K if for every intermediate field M the relative algebraic closure M∗ of M in L is normal (in the usual sense) over M. If L has the property that every M is uniquely determined by the automorphism group U(M) of L over M, then L is normal over K and, if the characteristic p=0, conversely; if p>0 the converse fails but a certain weaker conclusion is obtained. Various further results are found, and constructive aspects of normal extensions are explored. Some open questions are discussed, the most important one being: Do there exist transcendental normal extensions which are not algebraically closed?
So it seems that my question is a nearly 60 year-old problem which was considered but left unsolved by Krull. I am tempted to officially give up at this point, and perhaps write up an expository note informing (and warning?) contemporary readers about this circle of ideas. Comments, suggestions and/or advice would be most welcome...
P.S.: Thanks very much to M. Bonnot.
Best Answer
Edited (in view of new comments to this answer and the original question): I believe that Delgado missed the point that $M=Fix(Aut_M(L))$ isn't a formal consequence of $K=Fix(Aut_K(L))$ for algebraic extensions $K\subseteq M\subseteq L$. I took a closer look into the (master?) thesis. It doesn't claim to contain anything new, This work is a journey through the main ideas and sucessive [sic] generalizations of Galois Theory, towards the origins of Grothendieck’s theory of Dessins d’Enfants ... as the author puts it in his abstract.
The chapter on Galois theory just repeats well-known text book material, mostly without proofs. Considering the verbose character of this chapter, I'm sure the author would have said more than We immediately conclude that ... if there had been a novel aspect. To me it appears that he simply missed an essential aspect of Galois theory.
At any rate, from $K=Fix(Aut_K(L))$ alone we cannot conclude much, one somehow has to use the fact that $L/K$ is algebraic too as the following example shows: If $L=K(x)$ for a transcendental $x$, and if $K$ is infinite, then $K$ is the fixed field of $Aut_K(L)$, but for most rational functions $r(x)$ the extension $L/K(r(x))$ isn't Galois in either sense.
So if we want to show that $M=Fix(Aut_M(L))$ for an algebraic extension $L/K$ with $K=Fix(Aut_K(L))$, then I believe that one is automatically lead to the usual kind of arguments, which by the are also listed in this thesis.