Open Access
2013 Finitely presented exponential fields
Jonathan Kirby
Algebra Number Theory 7(4): 943-980 (2013). DOI: 10.2140/ant.2013.7.943


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.


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Jonathan Kirby. "Finitely presented exponential fields." Algebra Number Theory 7 (4) 943 - 980, 2013.


Received: 1 August 2011; Revised: 8 May 2012; Accepted: 12 May 2012; Published: 2013
First available in Project Euclid: 20 December 2017

zbMATH: 06207093
MathSciNet: MR3095232
Digital Object Identifier: 10.2140/ant.2013.7.943

Primary: 03C65
Secondary: 11J81

Keywords: exponential fields , pseudoexponentiation , Schanuel's conjecture , transcendence

Rights: Copyright © 2013 Mathematical Sciences Publishers

Vol.7 • No. 4 • 2013
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