Finitely Presented Exponential Fields

TL;DR

This paper develops the theory of finitely presented extensions in exponential fields, classifies strong extensions, constructs Zilber's pseudo-exponential fields, and explores their model-theoretic and transcendence properties.

Contribution

It introduces the concept of finitely presented extensions in exponential fields, classifies strong extensions, and provides an algebraic construction of Zilber's pseudo-exponential fields.

Findings

Zilber's fields are not model-complete.

Finitely generated strong extensions are finitely presented.

Schanuel's conjecture explains transcendence questions.

Abstract

The algebra of exponential fields and their extensions is developed. The focus is on ELA-fields, which are algebraically closed with a surjective exponential map. In this context, finitely presented extensions are defined, it is shown that finitely generated strong extensions are finitely presented, and these extensions are classified. An algebraic construction is given of Zilber's pseudo-exponential fields. As applications of the general results and methods of the paper, it is shown that Zilber's fields are not model-complete, answering a question of Macintyre, and a precise statement is given explaining how Schanuel's conjecture answers all transcendence questions about exponentials and logarithms. Connections with the Kontsevich-Zagier, Grothendieck, and Andr\'e transcendence conjectures on periods are discussed, and finally some open problems are suggested.