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we study a higher dimensional Lehmer problem, or alternatively the Lehmer problem for a power of the multiplicative group. More precisely, if α1, . . . , αn are multiplicatively independent algebraic numbers, we provide a lower bound for the product of the heights of the αi’s in terms of the degree D of the number field generated by the αi’s. This enables us to study the successive minima for the height function in a given number field. Our bound is a generalisation of an earlier result of Dobrowolski 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.

Le problème de Lehmer en dimension supérieure