Algebraic Independence

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


Liouville’s inequality which has been used many times so far (namely Lemma 2.1) involves polynomial approximations: a tuple θ = (θ l,..., θ m ) of complex numbers for which there are polynomials f ∈ ℤ[X1,..., X m ] such that |f ( θ)| is small but not zero contains at least one transcendental element. The converse is also true, and this yields a transcendence criterion. A similar statement holds for algebraic approximations to a complex number: a number θ ∈ ℂ is transcendental if and only if there are algebraic numbers γ such that |θ − γ|is small but not zero. One deduces that numbers θ 1,..., θ m belonging to a field of transcendence degree 1 admit good simultaneous approximations by algebraic numbers γ l,..., γ m , where the quality of the approximation, namely the number max1≤ i m |θ i γ i |, is controlled in terms of the degree [ℚ(γ 1,..., γ m ): ℚ].


Complex Number Algebraic Number Simultaneous Approximation Transcendence Degree Algebraic Independence 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

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

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