On Denominators of Algebraic Numbers and Integer Polynomials Original Research Article

LetA(x)=adxd+…+a0be the minimal polynomial ofαover image. Recall that the denominator ofα, denoted den(α), is defined as the least positive integernfor whichnαis an algebraic integer. It is well known that den(α)midad. In this paper we study the density of algebraic numbersαof fixed degreedsuch that den(α)=ad. We show that this density is given by[formula]Note that the above density approaches 1/ζ(3) asd→∞. As a result, we show, loosely speaking, that the chance that an algebraic numberαsatisfies den(α)=adis 1/ζ(3). In order to prove these results we introduce the concept of the denominator of an integer polynomialA. Several formulas for computing denominators of integer polynomials are derived.