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

In this paper, we study some special subfields of CC called Mahler fields. These fields are generated over QQ 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 UmUm-number is a UmUm-number. Using this result, we give another proof of the fact that the set of UmUm-numbers are non-empty for each m≥1m≥1. The famous Schanuel's conjecture states that, for any QQ-linearly independent complex numbers ξ1,…,ξnξ1,…,ξn, the transcendence degree of the field Q(ξ1,…,ξn)Q(ξ1,…,ξn) over QQ is at least n  . Here, we prove that for any QQ-linearly independent complex numbers ξ1,…,ξnξ1,…,ξn, there exist uncountably many U-numbers c   such that the transcendence degree of the field Q(cξ1,…,cξn,ecξ1,…,ecξn)Q(cξ1,…,cξn,ecξ1,…,ecξn) over QQ is at least n.