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Finitely Presented Exponential Fields

Authors
  • Kirby, Jonathan
Type
Published Article
Publication Date
Aug 01, 2011
Submission Date
Dec 20, 2009
Identifiers
DOI: 10.2140/ant.2013.7.943
Source
arXiv
License
Yellow
External links

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 Andr\'e transcendence conjectures on periods are discussed, and finally some open problems are suggested.

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