The Hermite-Joubert problem and a conjecture of Brassil-Reichstein

TL;DR

This paper proves that Hermite's theorem does not hold for certain integers expressed as sums of three powers of 3, confirming a conjecture, and extends results to the relative Hermite-Joubert problem over specific fields.

Contribution

It confirms Brassil and Reichstein's conjecture and advances understanding of the Hermite-Joubert problem over finitely generated fields of characteristic zero.

Findings

Hermite theorem fails for integers of the form 3^{k_1}+3^{k_2}+3^{k_3}

Confirmed a conjecture of Brassil and Reichstein

Extended results to the relative Hermite-Joubert problem over certain fields

Abstract

We show that Hermite theorem fails for every integer $n$ of the form $3^{k_1}+3^{k_2}+3^{k_3}$ with integers $k_1>k_2>k_3\geq 0$. This confirms a conjecture of Brassil and Reichstein. We also obtain new results for the relative Hermite-Joubert problem over a finitely generated field of characteristic $0$.