Finitely Presented Exponential Fields

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 André transcendence conjectures on periods are discussed, and finally some open problems are suggested.

39 pages; v4 Minor changes and an improved discussion of the connections with transcendence theory

References in corpus (6)

• On Quasiminimal Excellent Classes
• Exponential algebraicity in exponential fields
• The theory of the exponential differential equations of semiabelian varieties
• Covers of Multiplicative Groups of Algebraically Closed Fields of Arbitrary Characteristic
• The algebraic numbers definable in various exponential fields
• Generic Automorphisms and Green Fields

Cited by in corpus (9)

• Exponentially Closed Fields and the Conjecture on Intersections with Tori
• Pseudo-exponential maps, variants, and quasiminimality
• A note on the axioms for Zilber's pseudo-exponential fields
• Existentially Closed Exponential Fields
• A weak version of the Strong Exponential Closure
• A pseudoexponentiation-like structure on the algebraic numbers
• Independence relations for exponential fields
• Quasiminimality of complex powers
• Algebraic types in Zilber's exponential field