It is well known that a differential field K of characteristic 0 is contained in a differential field which is differentially closed$^1$ and has the property that it K-embeds in every differentially closed field containing K. Such a field is called a differential closure of K, and it is unique up to K-isomorphism. The proof of uniqueness uses fundamentally the uniqueness of prime models in totally transcendental theories.

One can ask the same question about difference fields: do they have a difference closure, and is it unique? The immediate answer to both these questions is no, for trivial reasons, and leads to impose some natural conditions on the difference field K. But algebraic examples show that these conditions are not sufficient.

As with differentially closed fields, model theory comes in and suggests natural strengthenings of the notion of difference closure (allowing for

instance countable systems of difference equations). It turns out that at least in characteristic 0, these difference closures exist and are

unique, under some natural assumption on the difference field K.

This result can then be used to show that certain automorphism groups are simple (joint with Blossier, Hardouin and Martin-Pizarro).

$^1$: Differentially closed: every finite system of differential equations which has a solution in a differential field extension, already has a solution in the differential field. Similar definition for difference closed: every finite system of difference equations which has a solution in a difference field extension, already has a solution in the difference field.