Cubic Subfields of Exceptional Simple Jordan Algebras

Let E/k be a cubic field extension and J a simple exceptional Jordan algebra of degree 3 over k. Then £ is a reducing field of J if and only if E is isomorphic to a (maximal) subfield of some isotope of /. If k has characteristic not 2 or 3 and contains the third roots of unity then every simple exceptional Jordan division algebra of degree 3 over k contains a cyclic cubic subfield. Exceptional Jordan division algebras were first constructed by Albert [2, 3]. Tits gave two constructions which yield all finite-dimensional exceptional simple Jordan algebras over fields of characteristic not 2 [4]. Then McCrimmon [5, 6] gave a particularly elegant characteristic free formulation of these results. Recently Zelmanov [14] proved that any exceptional Jordan division algebra of characteristic not 2 is finite dimensional over its center. Finally, the two Tits constructions were treated in a unified fashion via the Tits process introduced in [8]. There remains the isomorphism problem. If the algebra is not a division algebra, by results of Albert, Jacobson and Springer, isomorphism is equivalent to the isometry of quadratic forms in 8 and 24 variables (see [11] for a characteristic free version). In the case of division algebras the answer is not known and the results of this note represent small steps in the direction of a solution. We recall a few useful results and definitions. Let A; be a field. By a cubic form with adjoint and base point over k we mean a triple (N, #, 1) consisting of a cubic form N on some k vector space V, a quadratic map # : V -» V and a distinguished point 1 e V such that the identities