A transcendence criterion over \(p\)-adic fields (Q1428969)

Let \(p\) be a prime number, \({\mathbb Q}_ p\) the field of \(p\)-adic numbers, \(\overline{{\mathbb Q}} _ p\) a fixed algebraic closure of \({\mathbb Q}_ p\), \({\mathbb C}_ p\) the \(p\)-adic completion of \(\overline{{\mathbb Q}}_ p\) and \({\mathcal O}_{{\mathbb C}_ p}\) the ring of integers of \({\mathbb C}_ p\). The theory of saturated distinguished chains of polynomials plays an important role in the problem of describing the structure of irreducible polynomials in one variable over a local field \(K\). For instance, one can use a saturated distinguished chain for a given element \(\alpha\) in a fixed algebraic closure \(\overline{K}\) of \(K\) to construct an integral basis of \(K(\alpha)\) over \(K\). In the paper under review the use of saturated distinguished chains of polynomials allows the authors to characterize the elements \(t \in {\mathcal O}_{{\mathbb C}_ p}\) which are transcendental over \({\mathbb Q}_ p\). The main result is that if \(t\in {\mathcal O}_ {{\mathbb C}_ p}\) is transcendental over \({\mathbb Q}_ p\), then for any sequence \((t_ n)_ {n\in {\mathbb N}}\) in \({\mathbb C}_ p\) with \(\lim _ {n\to \infty} t_ n = t\) and any sequence of polynomials \((P_ n(X)) _ {n\in {\mathbb N}}\) in \({\mathbb Z}_ p[X]\) such that \(\lim _ {n \to \infty} P_ n (t_ n)=0\), we have \(\lim _ {n \to \infty} P'_ n (t_ n)=0\). Taking \(t_ n = t\) it is obtained that \(t\in {\mathcal O} _ {{\mathbb C}_ p}\) is transcendental over \({\mathbb Q}_ p\) if and only if for any sequence of polynomials \((P_ n(X))_ {n\in {\mathbb N}}\) in \({\mathbb Z}_ p[X]\) such that \(\lim _ {n \to \infty} P_ n (t)=0\), we have \(\lim _ {n \to \infty} P'_ n (t)=0\). It is also obtained that if \((\alpha _ n)_ {\alpha \in {\mathbb N}}\) is a Cauchy sequence of elements in \({\mathcal O}_ {{\overline{\mathbb Q}}_ p}\) and \(f_ {\alpha_ n}(x)\) denote the irreducible polynomial of \(\alpha _ n\) over \({\mathbb Q}_ p\) and either the sequence \((f'_{\alpha_ n}(\alpha _ n))_ {n\in {\mathbb N}}\) is not a Cauchy sequence, or this sequence is Cauchy but it does not converge to \(0\), then \(\lim_{n \to \infty} \alpha_ n\in {\mathbb C}_ p\) is algebraic over \({\mathbb Q}_ p\). The paper ends with an application concerning elements \(\beta \in {\mathcal O}_ {{\overline{\mathbb Q}}_ p}\) for which the differential \(d\beta\) vanishes.