Abstract
Using a work of Diaz concerning algebraic independence of certain sequences of numbers, we prove that if K â â is a field of finite transcendence degree over the rationals, then every weakly o-minimal expansion of (K,â€,+,·) is polynomially bounded. In the special case where K is the field of all real algebraic numbers, we give a proof which makes use of a much weaker result from transcendental number theory, namely, the Gelfond-Schneider theorem. Apart from this we make a couple of observations concerning weakly o-minimal expansions of arbitrary ordered fields of finite transcendence degree over the rationals. The strongest result we prove says that if K is a field of finite transcendence degree over the rationals, then all weakly o-minimal non-valuational expansions of (K,â€,+,·) are power bounded.
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