Approximation of a \(p\)-adic number by algebraic numbers (Q2781456)

In previous papers, \textit{Y. Bugeaud} and the author [Acta Arith. 93, 77-86 (2000; Zbl 0948.11029)] and the author [Monatsh. Math. 132, 169-176 (2001; Zbl 0976.11032)] used the method of \textit{H. Davenport} and \textit{W. Schmidt} [Acta Arith. 15, 393-416 (1969; Zbl 0186.08603)] to prove results of approximation of a real number by algebraic integers or units. Here, by the same method, the author obtains analogous results in the \(p\)-adic case: NEWLINENEWLINENEWLINEFor any integer \(n\geq 5\) let \(d(n)\) be the integral part of \((n-1)/2\) and \(v(n)\) the integral part of \((n+1)/2\). If \(\xi\in \mathbb{Q}_p\) is not an algebraic number of degree \(\leq d(n)\), then there exist infinitely many algebraic numbers \(\alpha\in \mathbb{Q}_p\) of degree exactly \(n-1\) such that \(|\xi- \alpha|_p\ll H(\alpha)^{-v(n)}\), where \(H(\alpha)\) is the naïve height of the algebraic number \(\alpha\). If furthermore \(\xi\in \mathbb{Z}_p\), then there exist infinitely many algebraic integers \(\alpha\in \mathbb{Q}_p\) of degree exactly \(n\) such that \(|\xi- \alpha|_p\ll H(\alpha)^{-v(n)}\). Actually, for \(2\leq n\leq 4\) the result is more accurate.