Finitely presented exponential fields

Abstract

We develop the algebra of exponential fields and their extensions. The focus is on ELA-fields, which are algebraically closed with a surjective exponential map. In this context, we define finitely presented extensions, show that finitely generated strong extensions are finitely presented, and classify these extensions. We give an algebraic construction of Zilber’s pseudoexponential fields. As applications of the general results and methods of the paper, we show that Zilber’s fields are not model-complete, answering a question of Macintyre, and we give a precise statement explaining how Schanuel’s conjecture answers all transcendence questions about exponentials and logarithms. We discuss connections with the Kontsevich–Zagier, Grothendieck, and André transcendence conjectures on periods, and suggest open problems.