On the Mahler measure of an algebraic number. (Sur la mesure de Mahler d’un nombre algébrique.) (French) Zbl 0557.12001

Louboutin, Roland

C. R. Acad. Sci., Paris, Sér. I 296, 707-708 (1983).

If \(x\neq 0\) is an algebraic number of degree d with conjugates \(x_ 1=x\), \(x_ 2,...,x_ d\) then the Mahler measure of x is \(M(x)=a_ 0\) \(\prod_{1\leq i\leq d}\max (1,| x_ i|)\) where \(a_ 0\) is the leading coefficient of the minimal polynomial of x. The author, improving a result of Dobrowolski, shows that for every \(\epsilon >0\), there is an integer D(\(\epsilon)\) such that for every algebraic number x, which is not zero or a root of unity, of degree \(d>D(\epsilon)\) we have \(M(x)>1+(9/4-\epsilon)(\log \log d/\log d)^ 3\).

Reviewer: A.Pollington

MSC:

11R04	Algebraic numbers; rings of algebraic integers
11J68	Approximation to algebraic numbers

Keywords:

algebraic number; Mahler measure