Vol. 7, No. 4, 2013

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Finitely presented exponential fields

Jonathan Kirby

Vol. 7 (2013), No. 4, 943–980

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.

exponential fields, Schanuel's conjecture, pseudoexponentiation, transcendence
Mathematical Subject Classification 2010
Primary: 03C65
Secondary: 11J81
Received: 1 August 2011
Revised: 8 May 2012
Accepted: 12 May 2012
Published: 29 August 2013
Jonathan Kirby
School of Mathematics
University of East Anglia
Norwich Research Park
United Kingdom