Diophantine conditions in small divisors and transcendental number theory

We present analogies between Diophantine conditions appearing in the theory of Small Divisors and classical Transcendental Number Theory. Let K be a number field. Using Bertrand's postulate, we give a simple proof that e is transcendental over Liouville fields K(theta) where theta is a Liouville number with explicit very good rational approximations. The result extends to any Liouville field K(Theta) generated by a family Theta of Liouville numbers satisfying a Diophantine condition (the transcendence degree can be uncountable). This Diophantine condition is similar to the one appearing in Moser's theorem of simultanneous linearization of commuting holomorphic germs.