$p$-adic transcendental numbers

Abstract

Explicit sets of cardinality 2'o of p-adic numbers which are algebraically independent over Qp are constructed. Let Qp be the p-adic completion of Q for prime p. Let Qp be the algebraic closure of Qp, and Cp be its p-adic completion which is an algebraically closed field of cardinality 2O . Let Qunram be the maximum unramiQp fied extension field of Qp. Then Qpunra = QP (W), where W is the set of all roots of unity whose orders are prime to p. Let Cunram be the p-adic closure p of Qunram in C,. Koblitz [1] asked whether Cunram has uncountably infinite transcendence degree over Qp and Cp has uncountably infinite transcendence degree over Cunram. Lampert [2] answered that the transcendence degree of p Cunram over Qp is 2tO and the transcendence degree of Cp over C unram is 2No by constructing sets of algebraically independent numbers of cardinality 2NO. Here we will give more explicit examples of such sets which cannot be obtained by the method in [2]. Theorem. Let K be an intermediate field between Qp and C P. Let ca 1' C.am be in Cp and a,, ... 5 aXnI be algebraically independent over K. Suppose that for i = 1, ... ,m 1 there exist sequences {flik}k>1 in Cp converging to and sequence {Sk}k>i of finite subsets of Aut(CP/K({1,lk}l<I<,_l )) which satisfies (1) lim S,jI = oo and ac i a forany , cSk with f , k--oo (2) max la fliklp =0 (min l<'7 lp as k oo where we define the left-hand side of (2) to be 0 if m = 1. Then (xl, .. m are algebraically independent over K. To prove the theorem we need the following lemma which is proved in Koblitz [1]. Received by the editors March 13, 1989. 1980 Matheinatics Subject Classification (1985 Revision). Primary 1 1J6 1. ? 1990 American Mathematical Society 0002-9939/90 $1.00 + $.25 per page