Submission date: 20 December 2009.

Abstract:

The algebra of exponential fields and their extensions is developed. Finitely presented extensions are defined, and it is shown that every finitely generated kernel-preserving strong extension is finitely presented. An algebraic construction is given of Zilber's pseudo-exponential fields. As applications of the method, it is shown that Zilber's fields are not model-complete, and their model-theoretic definable closure and algebraic closure are described, answering questions of Macintyre. A precise statement of why Schanuel's conjecture answers all transcendence questions about exponentials and logarithms is given.

Mathematics Subject Classification:

Keywords and phrases: 03C65; 11J81

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