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On Denominators of Algebraic Numbers and Integer Polynomials

✍ Scribed by Steven Arno; M.L. Robinson; Ferell S. Wheeler

Publisher
Elsevier Science
Year
1996
Tongue
English
Weight
543 KB
Volume
57
Category
Article
ISSN
0022-314X

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✦ Synopsis

Let A(x)=a d x d + } } } +a 0 be the minimal polynomial of : over Z. Recall that the denominator of :, denoted den(:), is defined as the least positive integer n for which n: is an algebraic integer. It is well known that den(:)|a d . In this paper we study the density of algebraic numbers : of fixed degree d such that den(:)=a d . We show that this density is given by

Note that the above density approaches 1Γ‚(3) as d Γ„ . As a result, we show, loosely speaking, that the chance that an algebraic number : satisfies den(:)=a d is 1Γ‚(3). In order to prove these results we introduce the concept of the denominator of an integer polynomial A. Several formulas for computing denominators of integer polynomials are derived.

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