The algebraic numbers definable using exponentiation

In the complex field, the only numbers which are pointwise definable (fixed under all field automorphisms) are the rational numbers. If we add the exponential function, we can ask which algebraic numbers become definable. We can distinguish the two square roots of 2, but what about cube roots, or other algebraic numbers? We introduce the notion of real abelian numbers, and show that they are all pointwise definable. Under a strong conjecture, we can show these are the only definable complex algebraic numbers. This is joint work with Macintyre and Onshuus.