The Joly–Becker theorem for $*$–orderings

Klep, Igor; Velušček, Dejan

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

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 $n\ell$ pour un certain (puis tout) impair $\ell \in \mathbb{N}$. 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 $2$.

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 $n\ell$ for some (resp. all) odd $\ell \in \mathbb{N}$. 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 $2$

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