Resumen de Minimal Mahler measures for generators of some fields

Artūras Dubickas

• We prove that for each odd integer d 3 there are infinitely many number fields K of degree d such that each generator ˛ of K has Mahler measure greater than or equal to d d jKj dC1 d.2d2/ , where 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 " > 0, there exist infinitely many number fields K of degree d such that every algebraic integer generator ˛ of K has Mahler measure greater than .1 "/jKj 1=d . On the other hand, every such field K contains an algebraic integer generator ˛ with Mahler measure smaller that jKj 1=d . This generalizes the corresponding bounds recently established by Eldredge and Petersen for d D=3.