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.
I don't think I have seen the terminology "admissible isomorphism" being used in differential Galois theory, except for Kaplansky's book. I guess in E. Kolchin's work everything is assumed to lie in a universal differential extension, and therefore he never makes the distinction.
Your observation about Picard-Vessiot extensions is right, and I don't think one needs the notion of admissibility to develop ordinary Picard-Vessiot theory, which is a theory based on the "equation" approach. In fact the efforts done at the time were focused on developing a Galois theory of differential fields that wasn't necessarily associated to differential equations (but it had to be a generalization of PV of course). However, many problems arise when one takes this "extension" approach, in fact finding the right notion of a normal extension is not easy. Classically a field extension $M$ over $K$ is normal if every isomorphism into some extension field of $M$ is an automorphism. However the equivalent statement for differential algebra implies that $M$ is algebraic over $K$ and that is too strong (in fact this is one of the main reasons why one has to allow admissible isomorphisms). Here are two early approaches to normality:
$M$ is weakly normal if $K$ is the fixed field of the set of all differential automorphisms of $M$ over $K$.
Apparently this definition wasn't very fruitful, and not much could be proven. The next step was the following definition:
$M$ is normal over $K$ if it is weakly normal over all intermediate differential fields.
This wasn't bad and Kolchin could prove that the map $L\to Gal(M/L)$ where $K\subset L\subset M$ bijects onto a certain subset of subgroups of $Gal (M/K)$. However the characterization of these subsets was an open question (Kolchin referred to it as a blemish). The property he was missing was already there in the theory of equations, as the existence of a superposition formula (that every solution is some differential rational function of the fundamental solutions and some constants). The relevant section in Kaplansky's book is sec 21. Now an admissible isomorphism of $M$ over $K$ is a differential isomorphism, fixing $K$ element wise, of $M$ onto a subfield of a given larger differential field $N$. Thus, an admissible isomorphism $\sigma$ let's you consider the compositum $M\cdot \sigma(M)$ which is crucial to translating a superposition principle to field extensions. Indeed, if one denotes $C(\sigma)$ to be the field of constants of $M\cdot \sigma(M)$, then Kolchin defined an admissible isomorphism $\sigma$ to be strong if it is the identity on the field of constants of $M$ and satisfies
$$M\cdot C(\sigma)=M\cdot \sigma(M)=\sigma (M)\cdot C(\sigma)$$
This was the right interpretation of what was happening in the PV case and so a strongly normal extension $M$ over $K$ was defined as an extension where $M$ is finitely generated over $K$ as a differentiable field, and every admissible isomorphism of $M$ over $K$ is strong. Now the theory became more complete. $Gal(M/K)$ may be identified with an algebraic group and there is a bijection between the intermediate fields and its closed subgroups. Now this incorporates finite normal extensions (when $Gal(M/K)$ is finite), Picard-Vessiot extensions (when $Gal(M/K)$ is linear) or Weierstrass extensions (when $Gal(M/K)$ is isomorphic to an elliptic curve).
For a better exposition of this, see if you can find "Algebraic Groups and Galois Theory in the Work of Ellis R. Kolchin" by Armand Borel.
Best Answer
The theory of differential Galois theory is used, but in algebraic, not differential geometry, under the name of D-modules. A D-module is an object that is somewhat more complicated than a representation of the differential Galois group, in the same way that a sheaf is a more complicated than just a Galois representation, but I think it is cut from the same cloth. A D-module describes not just the solutions of a differential equation but also how they behave at singularities.
D-modules are used in many different algebraic geometry situations.
While differential Galois theory may seem analytic it is actually much more algebraic. For instance, in analysis and differential geometry you tend to care how large things are, while in algebra you don't, and differential Galois theory says nothing about size. In algebra you hope for exact solutions, while in analysis approximate solutions are usually good enough, and differential Galois theory good for describing exact solutions. In differential geometry you often have great freedom in gluing together local pieces to get a global structure, where in algebra local pieces are rigid and hard to glue together, and differential Galois theory describes rigid structures where one tiny piece controls everything.