Anote on elimination of imaginaries for pairs of fields

Abstract

We show that, if $B$ is a field and $A \neq B$ is any algebraically closed  field extension of $B$, then the theory of $(A,B)$ eliminates imaginaries if  and only if $A$ is a finite extension of $B$ (and so if and only if $(A,B)$  is elementary equivalent to $ ( \mathbb{C}, \mathbb{R})$).