Summary: We study a higher-dimensional Lehmer problem, or alternatively the Lehmer problem for a power of the multiplicative group. More precisely, if are multiplicatively independent algebraic numbers, we provide a lower bound for the product of the heights of the ’s in terms of the degree of the number field generated by the ’s. This enables us to study the successive minima for the height function in a given number field. Our bound is a generalization of an earlier result of E. Dobrowolski [Acta Arith. 34, 391–401 (1979; Zbl 0416.12001)] and is best possible up to a power of . This, in particular, shows that the Lehmer problem is true for number fields having a ‘small’ Galois group.
The main result bases on two theorems, first an analogue of Dobrowolski’s key lemma and second a version of Philippon’s zero estimates with multiplication.