Unit Reducible Fields and Perfect Unary Forms
Alar Leibak1 ; Christian Porter2 ; Cong Ling2
1 Tallinn University of Technology Estonia
2 Imperial College London United Kingdom
Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 3, pp. 867-895.

Dans cet article, nous introduisons la notion de réductibilité unitaire pour les corps de nombres, c’est-à-dire les corps de nombres dans lesquels toutes les formes unaires positives atteignent leur minimum non nul en une unité. De plus, nous étudions le lien entre la réductibilité unitaire et le nombre de classes d’homothétie de formes unaires parfaites pour un corps de nombres donné, et prouvons une conjecture ouverte sur le nombre de classes de formes unaires parfaites dans des corps quadratiques réels, énoncée par D. Yasaki.

In this paper, we introduce the notion of unit reducibility for number fields, that is, number fields in which all positive unary forms attain their nonzero minimum at a unit. Furthermore, we investigate the link between unit reducibility and the number of homothety classes of perfect unary forms for a given number field, and prove an open conjecture about the number of classes of perfect unary forms in real quadratic fields, stated by D. Yasaki.

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DOI : 10.5802/jtnb.1267
Classification : 11E12, 11H55
Mots clés : Perfect Forms, Reduction Theory, Quadratic Forms, Algebraic Number Theory
Alar Leibak 1 ; Christian Porter 2 ; Cong Ling 2

1 Tallinn University of Technology Estonia
2 Imperial College London United Kingdom
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Alar Leibak; Christian Porter; Cong Ling. Unit Reducible Fields and Perfect Unary Forms. Journal de théorie des nombres de Bordeaux, Tome 35 (2023) no. 3, pp. 867-895. doi : 10.5802/jtnb.1267. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1267/

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