TITLE: Why do we need admissible isomorphisms for differential Galois theory?
QUESTION [15 upvotes]: Background: In Kaplansky's Introduction to Differential Algebra, an isomorphism between differential fields $K, L$ is defined to be admissible if $K,L$ are contained in a larger differential field $M$.  This appears to be an integral part of the development in his book.   For instance, Kaplansky proves the theorem that if $K, L$ are contained in a differential field $M$ and $f: K \to L$ is a differential isomorphism, then there is an admissible isomorphism $g: M \to M'$ extending $f$, using various differential analogs of results from commutative algebra (e.g. that a radical ideal is an intersection of prime ideals), which aren't at all necessary in ordinary field theory (as far as I know, anyway).
I find the definition hard to understand partially because it's not arrow-theoretic; it would help to have a more categorical notion than thinking in terms of subsets.  Moreover, in the case of Picard-Vessiot extensions (which if I understand correctly is what differential Galois theory focuses on), any admissible isomorphism is an automorphism anyway.  
Some googling suggests that one wants to take the (differential?) compositum between a differential field and its image under an admissible isomorphism.  Also, I have a suspicion that the absence of an analog of the concept of algebraic closure may be relevant here, but I'm not at all sure.
Question: To what extent is the restriction to admissibility necessary or useful, and if there's no way to avoid it for differential fields, why can one develop ordinary Galois theory without mentioning it?
And if it's indeed unavoidable, is there any way to think of it categorically?

REPLY [6 votes]: Expanding on Gjergji Zaimi's answer, (1) an hypothesis of admissibility is unnecessary and (2) the right context for differential Galois theories based on properties of extensions of differential fields as opposed to differential equations is somewhat more general than the strongly normal differential Galois theory of Kolchin.
There is an analogue of the notion of an algebraic closure for differential fields, called a constrained closure by Kolchin and his school and a differential closure in the model theoretic literature.  
Specializing to characteristic zero, a differential closure $(L,\partial_1,\ldots,\partial_n)$ of a partial differential field $(K,\partial_1,\ldots,\partial_n)$ is a differential field extension having the property that $L$ is differentially closed which means that every finite system of differential equations over $L$ which has a solution in some differential field extension of $L$ already has a solution in $L$ and is universal for differentially closed field extensions of $K$ in the sense that if $K \hookrightarrow M$ is an embedding of $K$ into a differentially closed field, then there is an embedding of $L$ into $M$ over $K$.   The differential closure of a differential field $K$ is unique up to isomorphism over $K$, but unlike the algebraic closure it is not minimal.  That is, if $L$ is the differential closure of $K$, then there may be a differentially closed field $M$ containing $K$ and properly contained in $L$.
The more general differential Galois theory I mentioned above is explained in the paper Pillay, Anand Differential Galois theory. I.  Illinois J. Math.  42  (1998),  no. 4, 678--699 at least in the case of ordinary differential fields.  Unlike Kolchin's theory, it is possible to have differential algebraic groups (instead of merely algebraic groups) as Galois groups.