Trace on Cp

Let p be a prime number, Q(p) the field of p-adic numbers, (Q) over bar (p) a fixed algebraic closure of Q(p), and C-p the completion of (Q) over bar (p). For elements T is an element of C-p which satisfy a certain diophantine condition (*) we construct a power series F(T, Z) with coefficients in Q(p) and show that two elements T, U produce the same series F(T, Z) = F(U, Z) if and only if they are conjugate. We view the coefficient of Z in F(T, Z) as the trace of T. Further, we study F(T, Z) viewed as a rigid analytic function and prove that it is defined everywhere on C-p except on the set of conjugates of 1/T. The main result (Theorem 7.2) asserts that if {T-alpha}(alpha) is a family of elements of C-p which are non-conjugate, transcendental over Q(p), and satisfy condition (*) then the functions {F(T-alpha, Z)}(alpha) are algebraically independent over C-p(Z). In particular, if T is an element of C-p(Z) if and only if T is transcendental over Q(p). In proving these results we develop some additional machinery, to be also used in a forthcoming paper which continues the study of orbits of elements in C-p. (C) 2001 Academic Press.