Survey and additional properties on the transcendence order over $\mathbb{Q}_p$ in $\mathbb{C}_p$

Abstract

The paper is aimed at recalling the notion of transcendence order over $\mathbb{Q}_p$ and its main properties. Proofs are more detailed than in the paper published in Journal of Number Theory. The main results: the order always is ≥ 1 and we construct a number b that is of order 1 + ∈ for every ∈ > 0. If a is of order ≤t and if b is transcendental over $\mathbb{Q}_p$ but algebraic over $\mathbb{Q}_p$, then b is of order ≤ t too. Finally, numbers of infinite order are constructed.