On separable pairs of fields and beautiful pairs of separably closed valued fields

Abstract

We continue the study of beautiful pairs of henselian valued fields, initiated in joint work of the second author with Cubides Kovacsics and Ye. As our main result, we show that separably closed non-trivially valued fields of arbitrary imperfection degree admit beautiful pairs which are elementary pairs satisfying certain natural axioms. They satisfy beauty transfer, i.e., saturated models of their theory are again beautiful. The same holds for beautiful pairs associated to natural subclasses of all definable types, most notably for all definable types orthogonal to the value group and for all bounded definable types. This yields strict pro-definability of the space of the definable types from any of these classes, in the theory SCVF of separably closed non-trivially valued fields. We also show a separable analog of a classical result of Keisler's, namely that the first order theory of an infinite separable extension $L/K$ of fields - considered in the pairs language and where the large field $L$ is separably closed, is determined by the theory of the small field $K$ together with the imperfection degree of $L$ over $K$.