A higher-dimensional Lehmer problem

We study a higher dimensional Lehmer problem, or alternatively the Lehmer p roblem for a power of the multiplicative group. More precisely, if alpha(1) ,..., alpha(n) are multiplicatively independent algebraic numbers, we provi de a lower bound for the product of the heights of the alpha(i)'s in terms of the degree D of the number field generated by the alpha(i)'s. This enabl es us to study the successive minima for the height function in a given num ber field. Our bound is a generalisation of an earlier result of Dobrowolsk i and is best possible up to a power of log(D). This, in particular, shows that the Lehmer problem is true for number fields having a <<small>> Galois group.