We give four different independence relations on any exponential field. Each is a canonical independence relation on a suitable Abstract Elementary Class of exponential fields, showing that two of these are NSOP1-like and non-simple, a third is stable, and the fourth is the quasiminimal pregeometry of Zilber's exponential fields, previously known to be stable (and uncountably categorical). We also characterise the fourth independence relation in terms of the third, strong independence.

Abstract A careful exposition of Zilber's quasiminimal excellent classes and their categoricity is given, leading to two new results: the Lω1ω(Q)-definability assumption may be dropped, and each class is determined by its model of dimension ℵ0.

Abstract I prove a version of Schanuel's conjecture for Weierstrass equations in differential fields, answering a question of Zilber, and show that the linear independence condition in the statement cannot be relaxed.