Fields of Mahler's U-numbers and Schanuel's conjecture

Abstract

In this paper, we study some special subfields of C called Mahler fields. These fields are generated over Q by a set of Mahler's U-numbers having approximation in a fixed algebraic number field. We completely classify their finite extensions. We provide a necessary condition for the non-zero polynomial image of a Um-number is a Um-number. Using this result, we give another proof of the fact that the set of Um-numbers are non-empty for each m≥1. The famous Schanuel's conjecture states that, for any Q-linearly independent complex numbers ξ1,…,ξn, the transcendence degree of the field Q(ξ1,…,ξn) over Q is at least n. Here, we prove that for any Q-linearly independent complex numbers ξ1,…,ξn, there exist uncountably many U-numbers c such that the transcendence degree of the field Q(cξ1,…,cξn,ecξ1,…,ecξn) over Q is at least n.