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].
KeywordsComplex Number Algebraic Number Simultaneous Approximation Transcendence Degree Algebraic Independence
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