La forme trace d'une algèbre simple centrale de degré 4

[The trace form of a central simple algebra of degree 4]

By Markus Rost, Jean-Pierre Serre, and Jean-Pierre Tignol.

M. Rost, J-P. Serre, J.-P. Tignol, C. R. Acad. Sci. Paris, Ser. I 342 (2006), no. 2, 83-87.

Article: doi.org, Académie des sciences.

MR 2193651, Zbl 1110.16014.

Let F be a field of characteristic not 2 containing a square root i of -1.

Theorem: Let A be a central simple F-algebra of degree 4. The trace form q_A of A is up to Witt equivalence the sum of a 2-fold Pfister form q₂ and a 4-fold Pfister form q₄. The Hasse-Witt invariant of q₂ in the Brauer group of F equals twice the class of A. The form q₄ is hyperbolic if and only if A is cyclic.

The exterior powers of q_A are determined. If there is an involution t on A with t(i)=-i, the theorem "descents" with respect to t. A major observation is a composition law for certain direct summands of q_A.

Full text (version of October 30, 2005): [pdf]

The case of characteristic 2 is considered in:

La forme seconde trace d'une algèbre simple centrale de degré 4 de caractéristique 2

[The second trace form of a central simple algebra of degree 4 of characteristic 2]

J.-P. Tignol, C. R. Acad. Sci. Paris, Ser. I 342 (2006), no. 2, 89-92.

Article: doi.org, Académie des sciences.

MR 2193652, Zbl 1110.16015.

Full text (version of November, 2005): [pdf]

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