Explicit lower bounds for the height in Galois extensions of number fields
Jonathan Jenvrin1
1 100 Rue des Mathématiques, 38610 Gières, France
Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 2, pp. 469-477

Amoroso and Masser proved that for every real $\epsilon > 0$, there exists a constant $c(\epsilon )>0$, with the property that, for every algebraic number $\alpha $ such that $\mathbb{Q}(\alpha )/\mathbb{Q}$ is a Galois extension, the height of $\alpha $ is either 0 or at least $c(\epsilon ) [\mathbb{Q}(\alpha ):\mathbb{Q}]^{-\epsilon }$. In the present article, we establish an explicit version of the aforementioned theorem.

Amoroso et Masser ont prouvé que, pour tout réel $\epsilon > 0$, il existe une constante $c(\epsilon ) > 0$, avec la propriété suivante : pour tout nombre algébrique $\alpha $ tel que $\mathbb{Q}(\alpha )/\mathbb{Q}$ est une extension galoisienne, la hauteur de $\alpha $ est soit nulle, soit au moins $c(\epsilon ) [\mathbb{Q}(\alpha ) : \mathbb{Q}]^{-\epsilon }$. Dans le présent article, nous établissons une version explicite de ce théorème.

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DOI : 10.5802/jtnb.1329
Classification : 11G50
Keywords: Height bounds, Galois extensions, Lehmer’s problem

Jonathan Jenvrin 1

1 100 Rue des Mathématiques, 38610 Gières, France
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Jonathan Jenvrin. Explicit lower bounds for the height in Galois extensions of number fields. Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 2, pp. 469-477. doi: 10.5802/jtnb.1329

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