Résume | In 1992 Lascar proved that the group of field automorphisms of the complex numbers which fix pointwise the algebraic closure of the rationals is simple, assuming the continuum hypothesis. His proof used strongly the topological features of the group of automorphisms of a countable structure, as a Polish group.
In 1997 Lascar gave a different proof of the above, without assuming the continuum hypothesis. The new proof needed just the stability of the theory of the field of complex numbers (and particularly stationarity of types as a way to merge two elementary maps) as well as the fact that the field of complex numbers is saturated in its own cardinality. In a recent preprint with T. Blossier, Z. Chatzidakis and C. Hardouin, we have adapted a proof of Lascar to show that certain groups of automorphisms of various theories of fields with operators are simple. It particularly applies to the theory of difference closed fields, which is simple and hence has possibly no models which are saturated in their uncountable cardinality. |