Heights of Algebraic Numbers

Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 326)


A nonzero rational integer has absolute value at least 1. A nonzero rational number has absolute value at least the inverse of any denominator. Liouville’s inequality (§ 3.5) is an extension of these estimates and provides a lower bound for the absolute value of any nonzero algebraic number. More specifically, if we are given finitely many (fixed) algebraic numbers γ l,...,γ t , and a polynomial P ∈ ℤ[X1,...,X t ] which does not vanish at the point (γ l,...,γ t ) then we can estimate from below |P(γ l,...,γ t )|. The lower bound will depend upon the degrees of P with respect to each of the X i ’s, the absolute values of its coefficients as well as some measure of the γ i ’s.


Number Field Algebraic Number Minimal Polynomial Product Formula Algebraic Integer 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  1. 1.Institut de Mathématiques de JussieuUniversité Pierre et Marie Curie (Paris VI)Paris Cedex 05France

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