Unit Reducible Fields and Perfect Unary Forms II. Cyclotomic Fields
Christian Porter1  ; Cong Ling1 ; Piero Sarti1 ; Alar Leibak2
1 Imperial College London, United Kingdom
2 Tallinn University of Technology, Estonia
Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 2, pp. 443-456

In this paper, we continue the study of unit reducible fields as introduced in [10] for the special case of cyclotomic fields. Specifically, we deduce that the cyclotomic fields of conductors $3,4,5,7,8,9,12,15$ are all unit reducible, and show that any cyclotomic field of conductor $N$ is not unit reducible if $2^4, 3^3, 5^2, 7^2, 11^2$ or any prime $p \ge 13$ divide $N$, meaning the unit reducible cyclotomic fields are finite in number. Finally, if $a$ is a totally positive element of a cyclotomic field, we show that for all equivalent $a^\prime $, the discrepancy between $\mathrm{Tr}_{K/\mathbb{Q}}(a^\prime )$ and the shortest nonzero element of the quadratic form $\mathrm{Tr}_{K/\mathbb{Q}}(axx^*)$ where $x$ is taken from the ring of integers tends to infinity as the conductor $N$ goes to infinity.

Disclaimer:

Due to an error during submission not caught in proofs, this article was initially published with the wrong title “Unit Reducible Fields and Perfect Unary Forms”.

Dans cet article, nous poursuivons l’étude des corps réductibles à l’unité, telle qu’introduite dans [10] pour le cas particulier des corps cyclotomiques. Plus précisément, nous en déduisons que les corps cyclotomiques des conducteurs $3,4,5,7,8,9,12,15$ sont tous réductibles à l’unité, et montrons que tout corps cyclotomique de conducteur $N$ n’est pas réductible à l’unité si $2^4$, $3^3$, $5^2, 7^2, 11^2$ ou tout nombre premier $p \ge 13$ divise $N$, ce qui signifie que les corps cyclotomiques réductibles à l’unité sont en nombre fini. Enfin, si $a$ est un élément totalement positif d’un corps cyclotomique, nous montrons que pour tout $a^\prime $ équivalent, l’écart entre $\mathrm{Tr}_{K/\mathbb{Q}}(a^\prime )$ et le plus court élément non nul de la forme quadratique $\mathrm{Tr}_{K/\mathbb{Q}}(axx^*)$ où $x$ est tiré de l’anneau des entiers tend vers l’infini lorsque le conducteur $N$ tend vers l’infini.

Avertissement :

En raison d’une erreur dans la soumission qui n’a pas été détectée lors de la mise en page, cet article a d’abord été publié sous le titre erroné « Unit Reducible Fields and Perfect Unary Forms ».

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/jtnb.1327
Classification : 11H55, 11H50
Keywords: Reduction Theory, Quadratic Forms, Algebraic Number Theory

Christian Porter 1 ; Cong Ling 1 ; Piero Sarti 1 ; Alar Leibak 2

1 Imperial College London, United Kingdom
2 Tallinn University of Technology, Estonia
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Christian Porter; Cong Ling; Piero Sarti; Alar Leibak. Unit Reducible Fields and Perfect Unary Forms II. Cyclotomic Fields. Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 2, pp. 443-456. doi: 10.5802/jtnb.1327

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