Let k be a field of characteristic different from 2 containing a primitive 4-th root of unity. We show that the trace quadratic form of any central simple k-algebra A of degree 4 decomposes in the Witt group of k as the sum of a 2-fold Pfister form and a 4-fold Pfister form which are uniquely determined by A. The form is the norm form of the quaternion algebra Brauer-equivalent to , and is hyperbolic if and only if A is a symbol algebra.
Soit k un corps de caractéristique ≠2 dans lequel −1 est un carré, et soit A une algèbre simple centrale sur k de degré 4. La forme trace de A s'écrit de façon unique comme une somme (au sens de Witt) , où (resp. ) est une 2-forme de Pfister (resp. une 4-forme de Pfister). On a si et seulement si A est cyclique, et si et seulement si dans .
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@article{CRMATH_2006__342_2_83_0, author = {Rost, Markus and Serre, Jean-Pierre and Tignol, Jean-Pierre}, title = {La forme trace d'une alg\`ebre simple centrale de degr\'e 4}, journal = {Comptes Rendus. Math\'ematique}, pages = {83--87}, publisher = {Elsevier}, volume = {342}, number = {2}, year = {2006}, doi = {10.1016/j.crma.2005.11.002}, language = {fr}, url = {http://archive.numdam.org/articles/10.1016/j.crma.2005.11.002/} }
TY - JOUR AU - Rost, Markus AU - Serre, Jean-Pierre AU - Tignol, Jean-Pierre TI - La forme trace d'une algèbre simple centrale de degré 4 JO - Comptes Rendus. Mathématique PY - 2006 SP - 83 EP - 87 VL - 342 IS - 2 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.crma.2005.11.002/ DO - 10.1016/j.crma.2005.11.002 LA - fr ID - CRMATH_2006__342_2_83_0 ER -
%0 Journal Article %A Rost, Markus %A Serre, Jean-Pierre %A Tignol, Jean-Pierre %T La forme trace d'une algèbre simple centrale de degré 4 %J Comptes Rendus. Mathématique %D 2006 %P 83-87 %V 342 %N 2 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.crma.2005.11.002/ %R 10.1016/j.crma.2005.11.002 %G fr %F CRMATH_2006__342_2_83_0
Rost, Markus; Serre, Jean-Pierre; Tignol, Jean-Pierre. La forme trace d'une algèbre simple centrale de degré 4. Comptes Rendus. Mathématique, Volume 342 (2006) no. 2, pp. 83-87. doi : 10.1016/j.crma.2005.11.002. http://archive.numdam.org/articles/10.1016/j.crma.2005.11.002/
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