A note on the axioms for Zilber’s pseudo-exponential fields. (English) Zbl 1345.03070

Summary: We show that Zilber’s conjecture that complex exponentiation is isomorphic to his pseudo-exponentiation follows from the a priori simpler conjecture that they are elementarily equivalent. An analysis of the first-order types in pseudo-exponentiation leads to a description of the elementary embeddings, and the result that pseudo-exponential fields are precisely the models of their common first-order theory which are atomic over exponential transcendence bases. We also show that the class of all pseudo-exponential fields is an example of a nonfinitary abstract elementary class, answering a question of Kesälä and Baldwin.


03C60 Model-theoretic algebra
03C48 Abstract elementary classes and related topics
12L12 Model theory of fields
Full Text: DOI arXiv Euclid


[1] Hyttinen, T., and M. Kesälä, “Independence in finitary abstract elementary classes,” Annals of Pure and Applied Logic , vol. 143 (2006), pp. 103-38. · Zbl 1112.03026
[2] Kirby, J., “Exponential algebraicity in exponential fields,” Bulletin of the London Mathematical Society , vol. 42 (2010), pp. 879-90. · Zbl 1203.03050
[3] Kirby, J., “Finitely presented exponential fields,” to appear in Algebra and Number Theory , preprint, [math.LO].
[4] Kirby, J., and B. Zilber, “Exponential fields and atypical intersections,” preprint, [math.LO]. · Zbl 1315.03056
[5] Kueker, D. W., “Abstract elementary classes and infinitary logics,” Annals of Pure and Applied Logic , vol. 156 (2008), pp. 274-86. · Zbl 1155.03016
[6] Macintyre, A. J., “Exponential algebra,” pp. 191-210 in Logic and Algebra (Pontignano, 1994) , vol. 180 of Lecture Notes in Pure and Applied Mathematics , Dekker, New York, 1996.
[7] Zilber, B., “Pseudo-exponentiation on algebraically closed fields of characteristic zero,” Annals of Pure and Applied Logic , vol. 132 (2005), pp. 67-95. · Zbl 1076.03024
[8] Zilber, B., “Covers of the multiplicative group of an algebraically closed field of characteristic zero,” Journal of the London Mathematical Society (2) , vol. 74 (2006), pp. 41-58. · Zbl 1104.03030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.