Survey and additional properties on the transcendence order over $$\mathbb{Q}_p$$ in $$\mathbb{C}_p$$
Alain Escassut · P-Adic Numbers Ultrametric Analysis and Applications · 2015
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.