The generating degree of ℂ_p

The generating degree gdeg(A) of a topological commutative ring A with char A = 0 is the cardinality of the smallest subset M of A for which the subring ℤ[M] is dense in A. For a prime number p, ℂ_p denotes the topological completion of an algebraic closure of the field ℚ_p of p-adic numbers. We prove that gdeg(ℂ_p) = 1, i.e., there exists t in ℂ_p such that ℤ[t] is dense in ℂ_p. We also compute gdeg(A(U)) where A(U) is the ring of rigid analytic functions defined on a ball U in ℂ_p. If U is a closed ball then gdeg(A(U)) = 2 while if U is an open ball then gdeg(A(U)) is infinite. We show more generally that gdeg(A(U)) is finite for any affinoid U in P¹(ℂ_p) and gdeg(A(U)) is infinite for any wide open subset U of P¹(ℂ_p).