Geometric classification of imaginaries in pseudofinite fields

In [Pil07], Pillay showed that every imaginary in a proper pair of algebraically closed fields is interalgebraic with an imaginary of a certain algebraic form. In this paper, we establish an analogue for lovely pairs of pseudofinite fields. After naming countably many constants, we prove that every imaginary in lovely pairs of pseudofinite fields is interalgebraic with an imaginary of the same algebraic form as [Pil07] in the corresponding pairs of finite extensions of pseudofinite fields.