Structure Theorems for Jordan Algebras of Degree Three Over Fields of Arbitrary Characteristic‡#

Abstract

Classical results, like the construction of a 3-fold Pfister form attached to any central simple associative algebra of degree 3 with involution of the second kind [Haile, D. E., Knus, M.-A., Rost, M., Tignol, J.-P. (1996). Algebras of odd degree with involution, trace forms and dihedral extensions. Israel J. Math. 96(B):299–340], or the Skolem–Noether theorem for Albert algebras and their 9-dimensional separable subalgebras [Parimala, R., Sridharan, R., Thakur, M. L. (1998). A classification theorem for Albert algebras. Trans. Amer. Math. Soc. 350(3):1277–1284], which originally were derived only over fields of characteristic not 2 (or 3), are extended here to base fields of arbitrary characteristic. The methods we use are quite different from the ones originally employed and, in many cases, lead to expanded versions of the aforementioned results that continue to be valid in any characteristic.