A note on generators of number fields

TL;DR

This paper provides new upper bounds on the minimal height of generators for number fields, with specific results for fields with real embeddings and those satisfying GRH, advancing understanding of number field generators.

Contribution

It introduces upper bounds for the height of generators of number fields, including a conditional result under GRH, addressing a question by W. Ruppert.

Findings

Upper bounds for fields with real embeddings

Conditional bounds assuming GRH for Galois closure

Partial answer to Ruppert's question

Abstract

We establish upper bounds for the smallest height of a generator of a number field $k$ over the rational field $\Q$. Our first bound applies to all number fields $k$ having at least one real embedding. We also give a second conditional result for all number fields $k$ such that the Dedekind zeta-function associated to the Galois closure of $k/\Q$ satisfies GRH. This provides a partial answer to a question of W. Ruppert.