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Abstract: The algebra of exponential fields and their extensions is developed. Thefocus is on ELA-fields, which are algebraically closed with a surjectiveexponential map. In this context, finitely presented extensions are defined, itis shown that finitely generated strong extensions are finitely presented, andthese extensions are classified. An algebraic construction is given of Zilber-spseudo-exponential fields. As applications of the general results and methodsof the paper, it is shown that Zilber-s fields are not model-complete,answering a question of Macintyre, and a precise statement is given explaininghow Schanuel-s conjecture answers all transcendence questions aboutexponentials and logarithms. Connections with the Kontsevich-Zagier,Grothendieck, and Andr\-e transcendence conjectures on periods are discussed,and finally some open problems are suggested.

Autor: Jonathan Kirby


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