Minimal Mahler measures for generators of some fields

We prove that for each odd integer d ≥ 3 there are infinitely many number fields K of degree d such that each generator α of K has Mahler measure greater than or equal to d⁻ᵈ|Δ_K|d+1/d(2d-2) , where Δ_K is the discriminant of the field K . This, combined with an earlier result of Vaaler and Widmer for composite d , answers negatively a question of Ruppert raised in 1998 about ‘small’ algebraic generators for every d ≥ 3 . We also show that for each d ≥ 2 and any ε>0 , there exist infinitely many number fields K of degree d such that every algebraic integer generator α of K has Mahler measure greater than (1-ε)|Δ_K|^{{1}/{d}} . On the other hand, every such field K contains an algebraic integer generator α with Mahler measure smaller that |Δ_K|^{{1}/{d}} . This generalizes the corresponding bounds recently established by Eldredge and Petersen for d=3 .