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The second trace form of a central simple algebra of degree 4 of characteristic 2. (La forme seconde trace d’une algèbre simple centrale de degré 4 de caractéristique 2.) (French) Zbl 1110.16015

Summary: Let \(A\) be a central simple algebra of degree 4 over a field \(k\) of characteristic 2 and let \(q_A\) be the quadratic form on \(A\) given by the second coefficient of the reduced characteristic polynomial. We show that \(A\) uniquely determines a 2-fold Pfister form \(q_2\) and a 4-fold Pfister form \(q_4\) such that \(q_A=[1,1]+q_2+q_4\) in the Witt group of \(k\), where \([1,1]\) is the form \(x^2+xy+y^2\). The form \(q_2\) is the norm form of the quaternion algebra Brauer-equivalent to \(A\otimes_kA\), and \(q_4\) is hyperbolic if and only if \(A\) is cyclic.

MSC:

16K20 Finite-dimensional division rings
11E81 Algebraic theory of quadratic forms; Witt groups and rings
11E04 Quadratic forms over general fields
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References:

[1] Albert, A. A., Structure of Algebras (1961), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI, MR0123587 (23 #A912) Zbl 0023.19901
[2] Amitsur, S. A.; Saltman, D., Generic Abelian crossed products and \(p\)-algebras, J. Algebra, 51, 76-87 (1978), MR0491789 (58 #10988) Zbl 0391.13001 · Zbl 0391.13001
[3] Bergé, A.-M.; Martinet, J., Formes quadratiques et extensions en caractéristique 2, Ann. Inst. Fourier (Grenoble), 35, 57-77 (1985), MR0786534 (87a:11036) Zbl 0539.10018 · Zbl 0539.10018
[4] Berhuy, G.; Frings, C., On the second trace form of central simple algebras in characteristic two, Manuscripta Math., 106, 1-12 (2001), MR1860978 (2002i:16024) Zbl 1003.11015 · Zbl 1003.11015
[5] Kneser, M., Composition of quaternary quadratic forms, Compositio Math., 60, 133-150 (1986), MR0868134 (88a:11037) Zbl 0612.10015 · Zbl 0612.10015
[6] Mammone, P.; Tignol, J.-P.; Wadsworth, A., Fields of characteristic 2 with prescribed \(u\)-invariants, Math. Ann., 290, 109-128 (1991), MR1107665 (92g:11035) Zbl 0713.12002 · Zbl 0713.12002
[7] M. Rost, J.-P. Serre, J.-P. Tignol, La forme trace d’une algèbre simple centrale de degré 4, C. R. Acad. Sci. Paris, Ser. I 342 (2006), in this issue; M. Rost, J.-P. Serre, J.-P. Tignol, La forme trace d’une algèbre simple centrale de degré 4, C. R. Acad. Sci. Paris, Ser. I 342 (2006), in this issue · Zbl 1110.16014
[8] Tignol, J.-P., Produits croisés abéliens, J. Algebra, 70, 420-436 (1981), MR0623817 (84f:16026) Zbl 0473.16004 · Zbl 0473.16004
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