2002/12/16 by Benoit Arbour, Damien Roy
Mathematics · #math.NT #msc:11J13
published as Acta Arithmetica 111.1 (2004), 97-103 · 6 pages
arxiv created 2002/12/16 · arxiv updated 2009/11/30
We establish a criterion for a complex number to be algebraic over Q of degree at most two. It requires that, for any sufficiently large real number X, there exists a non-zero polynomial with integral coefficients, of degree at most two and height at most X, whose absolute value at that complex number is at most (1/4)X^-(3+sqrt5)/2. We show that the exponent (3+sqrt5)/2 in this condition is optimal, and deduce from this criterion a result of simultaneous approximation of a real number by conjugate algebraic numbers.