[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.
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 qA 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 qA 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 qA.
Full text (version of October 30, 2005): [pdf]
The case of characteristic 2 is considered in:
J.-P. Tignol, C. R. Acad. Sci. Paris, Ser. I 342 (2006), no. 2, 89-92.
Full text (version of November, 2005): [pdf]