## Abstract

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 P^{1}(ℂ_{p}) and gdeg(A(U)) is infinite for any wide open subset U of P^{1}(ℂ_{p}).

Original language | English (US) |
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Pages (from-to) | 3-11 |

Number of pages | 9 |

Journal | Canadian Mathematical Bulletin |

Volume | 44 |

Issue number | 1 |

DOIs | |

State | Published - Mar 2001 |

Externally published | Yes |

## ASJC Scopus subject areas

- Mathematics(all)