## The Approximation of Algebraic Numbers by Algebraic Numbers and the Theory of Transcendental Numbers |

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A. A. Markov absolute value alge algebraic differential equation algebraic equations algebraic field algebraic independence algebraic irrationalities algebraic numbers analytic functions Approximation algebraischer Zahlen approximation of algebraic arbitrarily small arithmetical nature bers Bessel function braic numbers C. R. Acad circle class of numbers D. D. Morduhai-Boltovskoi dence denominators different from zero diophantine equations elliptic function Euler existence finite number follows formulate Fourth Step function f(z Gel'fond Hermite inequality irrational irreducible irreducible polynomial Journ least common multiple Lemma Lindemann's theorem linear forms Liouville number Liouville's lower bounds Mahler Math measure of transcendence mutual transcendence N. I. Fel'dman nombre number 77 number of solutions number of zeros obtained proof of Lindemann's prove the transcendence rational field rational fractions rational integer coefficients rational number root second step Siegel step consisted sufficiently large solutions Taylor series teger theory of numbers theory of transcendental Third Step tion transcendental character transcendental numbers