A note on generators of number fields
Jeffrey D. Vaaler, Martin Widmer

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 over the rational field . Our first bound applies to all number fields having at least one real embedding. We also give a second conditional result for all number fields such that the Dedekind zeta-function associated to the Galois closure of satisfies GRH. This provides a partial answer to a question of W. Ruppert.
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Taxonomy
TopicsVietnamese History and Culture Studies · Algebraic Geometry and Number Theory · Analytic Number Theory Research
