Let p be a prime number, Q p the field of p-adic numbers, Q p a fixed algebraic closure of Q p , and C p the completion of Q p . For elements T # C p which satisfy a certain diophantine condition (V) 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 1ΓT. The main result (Theorem 7.2) asserts that if [T : ] : is a family of elements of C p which are non-conjugate, transcendental over Q p , and satisfy condition (V) then the functions [F(T : , Z)] : are algebraically independent over C p (Z). In particular, if T is an element of C p which satisfies condition (*), then F(T, Z) is transcendental over 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 .