Ruppert’s small-generator conjecture for number fields

Determine whether every number field L of degree n>1 satisfies the bound \(\delta(L)\ll_n \Delta_L^{1/(2n)}\), where \(\delta(L)\) denotes the minimum absolute multiplicative Weil height of a primitive generator of L and \(\Delta_L\) is the absolute discriminant of L.

Background

The paper defines $\delta(L)$ as the minimum Weil height among algebraic numbers $\alpha$ satisfying $L=\mathbb{Q}(\alpha)$. Ruppert asked whether this minimum is always bounded by a constant depending only on the degree times $\Delta_L^{1/(2n)}$.

The question is known affirmatively for quadratic fields, for fields with a real embedding, and under the Riemann hypothesis for the Dedekind zeta function of the normal closure. The paper proves the same bound under the weaker quasi-Riemann hypothesis assumption, but does not resolve the unconditional question for all number fields.