Abstract

We study the structures $$, where $K$ is an ordered field and $K^\mathrm{r}$ its real closure, in the language of ordered fields with an additional unary predicate for the subfield $K$. Two such structures $$ and $$ are not necessarily elementary equivalent when $K$ and $L$ are. But with some saturation assumption on $K$ and $L$, then the two structures become equivalent, and we give a description of the complete theory.