Definable Skolem Functions in Some Complete Structures

Let L expand the language of ordered fields and DCT C denote the L-theory of L-structures which are definably complete (namely, they have no any definable Dedekind gap) and type complete (namely, order types generated by the elements of the structure are complete). These complete structures have been recently investigated by Paolo Fornasiero and Hans Schoutens, including remarkable results. On the other hand, having definable Skolem functions for definable relations is a tame property which holds in some first order structures. In this note, we show that models of DCT C have definable Skolem functions. Also, we show that there is a type complete ordered field which is discrete definable complete but not Dedekind definable complete.