On \(\mathfrak {P}\)-adic continued fractions with extraneous denominators: some explicit finiteness results

Abstract

Let K be a number field. We show that, up to allowing a finite set of denominators in the partial quotients, it is possible to define algorithms for \(\mathfrak {P}\)-adic continued fractions satisfying the finiteness property on K for every prime ideal \(\mathfrak {P}\) of sufficiently large norm. This provides, in particular, a new algorithmic approach to the construction of division chains in number fields.