The Generating Degree of ℂp

Abstract

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