Primes in denominators of algebraic numbers

Abstract

Denote the set of algebraic numbers as [Formula: see text] and the set of algebraic integers as [Formula: see text]. For [Formula: see text], consider its irreducible polynomial in [Formula: see text], [Formula: see text]. Denote [Formula: see text]. Drungilas, Dubickas and Jankauskas show in a recent paper that [Formula: see text]. Given a number field [Formula: see text] and [Formula: see text], we show that there is a subset [Formula: see text], for which [Formula: see text]. We prove that [Formula: see text] is a principal ideal domain if and only if the primes in [Formula: see text] generate the class group of [Formula: see text]. We show that given [Formula: see text], we can find a finite set [Formula: see text], such that for every number field [Formula: see text], we have [Formula: see text]. We study how this set [Formula: see text] relates to the ring [Formula: see text] and the ideal [Formula: see text] of [Formula: see text]. We also show that [Formula: see text] satisfy [Formula: see text] if and only if [Formula: see text] for all number fields [Formula: see text].