Number fields without small generators

TL;DR

This paper demonstrates that infinitely many number fields of a given degree have primitive elements with large height, especially for composite degrees, providing a negative answer to Ruppert's question.

Contribution

It establishes the existence of infinitely many number fields with primitive elements of large height, addressing Ruppert's question for composite degrees and under certain conjectural assumptions.

Findings

Infinitely many number fields have primitive elements with large height.

Negative answer to Ruppert's question for composite degrees.

Conditional negative answer for all degrees greater than 3.

Abstract

Let $D>1$ be an integer, and let $b=b(D)>1$ be its smallest divisor. We show that there are infinitely many number fields of degree $D$ whose primitive elements all have relatively large height in terms of $b$, $D$ and the discriminant of the number field. This provides a negative answer to a questions of W. Ruppert from 1998 in the case when $D$ is composite. Conditional on a very weak form of a folk conjecture about the distribution of number fields, we negatively answer Ruppert's question for all $D>3$.