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

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