×

Totally decomposable quadratic pairs. (English) Zbl 1370.11051

The authors state about their result: “Algebras with quadratic pair associated to quadratic Pfister forms are tensor products of quaternion algebras with involution and a quaternion algebra with quadratic pair. One may ask whether all such totally decomposable quadratic pairs on a split central simple algebra are adjoint to a quadratic Pfister form. In characteristic different from two, where quadratic pairs are equivalent to orthogonal involutions, this is known as Pfister factor conjecture proved by K. J. Becher [Invent. Math. 173, No. 1, 1–6 (2008; Zbl 1226.11049)], which says that in this case a totally decomposable orthogonal involution on a split algebra is adjoint to a Pfister form. In this article we prove the corresponding result for quadratic pairs over fields of characteristic two.”

MSC:

11E39 Bilinear and Hermitian forms
11E81 Algebraic theory of quadratic forms; Witt groups and rings
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
16K20 Finite-dimensional division rings

Citations:

Zbl 1226.11049

References:

[1] Albert, A.: Structure of Algebras, American Mathematical Society Colloquium Publications, vol. 24. American Mathematical Society, New York (1968) · Zbl 0218.17010
[2] Bayer-Fluckiger, E., Parimala, R., Quéguiner-Mathieu, A.: Pfister involutions. Proc. Indian Acad. Sci. Math. Sci. 113, 365-377 (2003) · Zbl 1049.16011 · doi:10.1007/BF02829631
[3] Becher, K.J.: A proof of the Pfister Factor Conjecture. Invent. Math. 173, 1-6 (2008) · Zbl 1226.11049 · doi:10.1007/s00222-007-0107-5
[4] Becher, K.J., Dolphin, A.: Non-hyperbolic splitting of quadratic pairs. J. Algebra Appl. 14, 1550138 (2015) · Zbl 1326.11013 · doi:10.1142/S0219498815501388
[5] Berhuy, G., Frings, C., Tignol, J.-P.: Galois cohomology of the classical groups over imperfect fields. J. Pure Appl. Algebra 211, 307-341 (2007) · Zbl 1121.11035 · doi:10.1016/j.jpaa.2007.01.001
[6] Dolphin, A.: Metabolic involutions. J. Algebra 336(1), 286-300 (2011) · Zbl 1277.11030 · doi:10.1016/j.jalgebra.2011.02.024
[7] Dolphin, A.: Orthogonal Pfister involutions in characteristic two. J. Pure Appl. Algebra 218(10), 1900-1915 (2014) · Zbl 1297.11020 · doi:10.1016/j.jpaa.2014.02.013
[8] Draxl, P.K.: Skew Fields. London Mathematical Society Lecture Note Series, Vol. 81. Cambridge University Press (1983) · Zbl 0498.16015
[9] Elman, R., Karpenko, N., Merkurjev, A.: The Algebraic and Geometric Theory Quadratic Forms, Vol. 56 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence (2008) · Zbl 1165.11042 · doi:10.1090/coll/056
[10] Hoffmann, D., Laghribi, A.: Isotropy of quadratic forms over the function field of a quadric in characteristic 2. J. Algebra 295, 362-386 (2006) · Zbl 1138.11012 · doi:10.1016/j.jalgebra.2004.02.038
[11] Karpenko, N.: On isotropy of quadratic pair. In: Baeza, R., Chan, W.K., Hoffmann, D.W., Schulze-Pillot, R. (eds.) Quadratic Forms-Algebra, Arithmetic, and Geometry, Contemporary Mathematics, vol. 493, pp. 211-217. American Mathematics Society, Providence (2009) · Zbl 1193.11035
[12] Knus, M.-A., Merkurjev, A.S., Rost, M., Tignol, J.-P.: The Book of Involutions, vol. 44 of American Mathematical Society Colloquium Publications. American Mathematical Society, Providence (1998) · Zbl 0955.16001
[13] Mammone, P., Tignol, J.-P., Wadsworth, A.: Fields of characteristic \[22\] with prescribed \[u\] u-invariants. Math. Ann. 290, 109-128 (1991) · Zbl 0713.12002 · doi:10.1007/BF01459240
[14] Pierce, R.: Associative Algebras. Graduate Texts in Mathematics, Springer, New York (1982) · Zbl 0497.16001 · doi:10.1007/978-1-4757-0163-0
[15] Shapiro, D.: Compositions of Quadratic Forms, de Gruyter Expositions in Mathematics, vol. 33. Walter de Gruyter, Berlin (2000) · Zbl 0954.11011 · doi:10.1515/9783110824834
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.