The generating degree of ℂp

Victor Alexandru, Nicolae Popescu, Alexandru Zaharescu

Research output: Contribution to journalArticlepeer-review

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, ℂp denotes the topological completion of an algebraic closure of the field ℚp of p-adic numbers. We prove that gdeg(ℂp) = 1, i.e., there exists t in ℂp such that ℤ[t] is dense in ℂp. We also compute gdeg(A(U)) where A(U) is the ring of rigid analytic functions defined on a ball U in ℂp. 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(ℂp) and gdeg(A(U)) is infinite for any wide open subset U of P1(ℂp).

Original languageEnglish (US)
Pages (from-to)3-11
Number of pages9
JournalCanadian Mathematical Bulletin
Volume44
Issue number1
DOIs
StatePublished - Mar 2001
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)

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