Transcendence order over Qp in Cp

Let (K, ∥ · ∥) be a valued transcendence degree 1 extension of $\text{Q}$_p. An element x ∈ K transcendental over $\text{Q}$_p is said to have order ≤a (a > 0) if there exists C_x > 0 such that every polynomial P(X) ∈ $\text{Q}$_p X] satisfies

$\text{?}\text{log;}\text{(P(x))? ?}\text{log}\text{∥P∥+c}{}_{\mathrm{x}}\text{(}\text{deg}\text{P)}{}^{\mathrm{a}}$

when ∥ · ∥ is the Gauss norm on $\text{Q}$_pX]. No x ∈ $\text{C}$_p can have order ≤α if α < 1 but we construct some x ∈ $\text{C}$_p with order ≤ 1. Furthermore, we prove order ≤α is stable by algebraic extension.