The Joly–Becker theorem for $*$–orderings

Igor Klep; Dejan Velušček

doi:10.5802/afst.1177

The Joly–Becker theorem for

*

–orderings

Igor Klep ¹; Dejan Velušček ²

¹ Univerza v Ljubljani, Oddelek za matematiko Inštituta za matematiko, fiziko in mehaniko, Jadranska 19, SI–1111 Ljubljana, Slovenia
² University of Ljubljana, Faculty of Mathematics and Physics, Department of Mathematics, Jadranska 19, SI–1000 Ljubljana, Slovenia.

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 17 (2008) no. 1, pp. 81-92.

Abstract
Abstract

We prove the $*$ –version of the Joly–Becker theorem: a skew field admits a $*$ –ordering of level $n$ iff it admits a $*$ –ordering of level $n ℓ$ for some (resp. all) odd $ℓ \in ℕ$ . For skew fields with an imaginary unit and fields stronger results are given: a skew field with imaginary unit that admits a $*$ –ordering of higher level also admits a $*$ –ordering of level $1$ . Every field that admits a $*$ –ordering of higher level admits a $*$ –ordering of level $1$ or $2$

Nous démontrons la version involutive du théorème de Joly et Becker : une algèbre à division admet un ordre involutif de niveau $n$ si et seulement si elle admet un ordre involutif de niveau $n ℓ$ pour un certain (puis tout) impair $ℓ \in ℕ$ . Dans le cas d’une algèbre à division avec une unité imaginaire ou d’un corps commutatif, nous présentons des résultats plus forts : si une algèbre à division avec unité imaginaire admet un ordre involutif de niveau supérieur, elle admet aussi un ordre involutif de niveau $1$ . Tout corps admettant un ordre involutif de niveau supérieur admet un ordre involutif de niveau $1$ ou $2$ .

MR

DOI: 10.5802/afst.1177

Author's affiliations:

Igor Klep ¹; Dejan Velušček ²

¹ Univerza v Ljubljani, Oddelek za matematiko Inštituta za matematiko, fiziko in mehaniko, Jadranska 19, SI–1111 Ljubljana, Slovenia
² University of Ljubljana, Faculty of Mathematics and Physics, Department of Mathematics, Jadranska 19, SI–1000 Ljubljana, Slovenia.

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     author = {Igor Klep and Dejan Velu\v{s}\v{c}ek},
     title = {The {Joly{\textendash}Becker} theorem for $*${\textendash}orderings},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {81--92},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
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Igor Klep; Dejan Velušček. The Joly–Becker theorem for $*$–orderings. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 17 (2008) no. 1, pp. 81-92. doi : 10.5802/afst.1177. https://afst.centre-mersenne.org/articles/10.5802/afst.1177/

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