no. 15-16
Algebra
On chains in division algebras of degree 3
[Chaines d'éléments de Kummer dans les corps gauches de degré 3]

Présenté par : Jean-Pierre Serre

Haile, Darrell1 ; Kuo, Jung-Miao2 ; Tignol, Jean-Pierre3
1 Department of Mathematics, Indiana University, Bloomington, IN 47401, USA
2 Department of Mathematics, National Taiwan University, Taipei, Taiwan
3 Université catholique de Louvain, chemin du cyclotron, 2, B-1348 Louvain-la-Neuve, Belgium
Comptes Rendus. Mathématique, Tome 347 (2009) no. 15-16, pp. 849-852.

Soit D un corps gauche de degré 3 sur un corps contenant une racine cubique de l'unité. Nous donnons deux démonstrations d'un théorème de Rost établissant que deux éléments de Kummer quelconques de D peuvent être joints par une chaine de longueur 4.

Let D be a division algebra of degree 3 over a field containing a primitive cube root of unity. We give two proofs of a theorem of Rost asserting that any two Kummer elements in D can be connected by a chain of length 4.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2009.06.007
Haile, Darrell 1 ; Kuo, Jung-Miao 2 ; Tignol, Jean-Pierre 3

1 Department of Mathematics, Indiana University, Bloomington, IN 47401, USA
2 Department of Mathematics, National Taiwan University, Taipei, Taiwan
3 Université catholique de Louvain, chemin du cyclotron, 2, B-1348 Louvain-la-Neuve, Belgium 
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Haile, Darrell; Kuo, Jung-Miao; Tignol, Jean-Pierre. On chains in division algebras of degree 3. Comptes Rendus. Mathématique, Tome 347 (2009) no. 15-16, pp. 849-852. doi : 10.1016/j.crma.2009.06.007. http://www.numdam.org/articles/10.1016/j.crma.2009.06.007/

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[3] Knus, M.-A. et al. The Book of Involutions, Amer. Math. Soc., Providence, RI, 1998

[4] Rost, M. The chain lemma for Kummer elements of degree 3, C. R. Acad. Sci. Paris, Sér. I, Volume 328 (1999) no. 3, pp. 185-190

[5] Tits, J. Buildings of Spherical Type and Finite BN Pairs, Lecture Notes in Mathematics, Springer-Verlag, New York, 1974

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