Algebraic independence of elements from Cpover Qp, II
Peter Bundschuh, Владимир Григорьевич Чирский · Acta Arithmetica · 2004
Dedicated to Rob Tijdeman on the occasion of his sixtieth birthday1. Introduction, main results, applications.Let Q p be the p-adic completion of Q for a prime p. Denote by Z p the ring of p-adic integers, i.e. of those elements x ∈ Q p with |x| p ≤ 1.The unit group of Q p , i.e. the set of x ∈ Q p with |x| p = 1, will be denoted by U p .Let Q p be the algebraic closure of Q p and C p its p-adic completion, which is an algebraically closed complete field with a valuation uniquely extended from Q p .Whereas the questions of transcendence or algebraic independence of elements from Q p or even from C p over Q are rather well investigated, the corresponding question for C p over Q p has been studied in the past only occasionally.For a brief survey on what was published on this topic so far, we refer the reader to our recent paper [3].The main result there gives sufficient conditions for the algebraic independence over Q p of numbers from C p defined by infinite series of the form a k p r k , where (r k ) is a sequence of positive rational numbers and the coefficients a k are p-adic integers.In our present paper, we propose two new such criteria, where the hypotheses on the a k , r k are now slightly stronger.But, on the other hand, we no longer need, as in [3], conditions on determinants involving certain of the coefficients a occurring in the different series under consideration.Both of these criteria have the same appearance, typical in algebraic independence theory: Under appropriate assumptions on functions f 1 , . . ., f l and points α 1 , . . ., α m , the l • m numbers f λ (α µ ) from C p are algebraically independent over Q p .