Some remarks on relative Lehmer.
by
Francesco Amoroso
University of Caen
Coauthors: Umberto Zannier (Scuola Normale Superiore - Pisa - Italy)

Let K be a number field and let L be an abelian extension of K. Let x be a non-zero algebraic number in L which is not a root of unity. As a a very special case of the main result of a previous joint paper with U. Zannier, the Weil height of x is bounded from below by a positive function of K. The proof gives a function which naturally depends on the degree *and* on the discriminant of K.

Here we show that the height of x can be bounded from below by a positive function depending *only* of the degree of K over the rational field.

As a corollary, in a dihedral extension of the rational field, the height of a non zero algebraic number which is not a root of unity is bounded from below by an absolute positive constant.

Date received: April 20, 2008