Joubert's theorem fails in characteristic 2

Reichstein, Zinovy

doi:10.1016/j.crma.2014.08.004

Number theory/Algebra

Joubert's theorem fails in characteristic 2

Présenté par : Jean-Pierre Serre

Reichstein, Zinovy ¹

¹ Department of Mathematics, University of British Columbia, Vancouver, Canada

Comptes Rendus. Mathématique, Tome 352 (2014) no. 10, pp. 773-777.

Résumé
Abstract

Soit $L / K$ une extension de corps séparable de degré 6. En 1867, P. Joubert a démontré que, si la caractéristique de K est différente de 2, l'extension $L / K$ est engendrée par un élément dont le polynôme minimal est de la forme $t^{6} + a t^{4} + b t^{2} + c t + d$ , pour des éléments convenables $a, b, c, d \in K$ . Dans cette note, nous démontrons que ce théorème ne s'étend pas à la caractéristique 2.

Let $L / K$ be a separable field extension of degree 6. A 1867 theorem of P. Joubert asserts that if $char (K) \neq 2$ , then L is generated over K by an element whose minimal polynomial is of the form $t^{6} + a t^{4} + b t^{2} + c t + d$ for some $a, b, c, d \in K$ . We show that this theorem fails in characteristic 2.

Reçu le : 2014-06-24
Accepté le : 2014-08-07
Publié le : 2014-09-18

DOI : 10.1016/j.crma.2014.08.004

Affiliations des auteurs :

Reichstein, Zinovy ¹

¹ Department of Mathematics, University of British Columbia, Vancouver, Canada

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Reichstein, Zinovy. Joubert's theorem fails in characteristic 2. Comptes Rendus. Mathématique, Tome 352 (2014) no. 10, pp. 773-777. doi : 10.1016/j.crma.2014.08.004. http://www.numdam.org/articles/10.1016/j.crma.2014.08.004/

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