The generating degree of \(\mathbb{C}_p\) (Q2739245)

Given a topological commutative ring \(A\) of characteristic \(0\), the generating degree \(\text{gdeg}(A)\), is the cardinality of the smallest subset \(M\) of \(A\) such that \({\mathbb{Z}}[M]\) is dense in \(A\). Let \({\mathbb{C}} _p\) be the completion of the algebraic closure of the field of \(p\)-adic numbers \({\mathbb{Q}}_p\). The authors are mainly interested in \(\text{gdeg}( {\mathbb{C}}_p)\). The first main result is that for any closed subfield \(E\) of \({\mathbb{C}}_p\) we have that \(\text{gdeg}(E) = 1\). It is also shown that if \(E\) is a closed subring of \({\mathbb{C}}_p\) not contained in the ring of integers \({\mathcal O}_{{\mathbb{C}}_p}\), then \(E\) is a field. In contrast, we have that \(\text{gdeg}({\mathcal O}_{{\mathbb{C}}_p}) = \infty\). In the last section the authors consider the rings \(A(U)\) of rigid analytic functions defined on an open subset \(U\) of \({\mathbb{C}}_p\). Similarly for open and closed balls in \({\mathbb{P}}^1({\mathbb{C}} _p)\). It is proved that if \(U\) is an affinoid, \(\text{gdeg( A(U))} < \infty\) and if \(U\) is a ``wide open set'' then \(\text{gdeg}(A(U)) = \infty\).