On the invariants mod 2 of Albert algebras
For a field \(\mathbb{K}\) of characteristic different from 2, Jordan algebras over \(\mathbb{K}\) are defined as those commutative algebras over \(\mathbb{K}\) satisfying the ``Jordan identity'' \(x^2 \circ (y \circ x) = (x^2 \circ y) \circ x\). Jordan algebras which cannot be seen as subspaces of associative algebras closed under the ``Jordan product'' \(x \circ y := {1\over 2} (xy + yx)\) are called exceptional. It is known that central simple exceptional Jordan algebras over \(\mathbb{K}\) (in short, Albert algebras over \(\mathbb{K}\)) are finite-dimensional. More precisely, the dimensions of Albert algebras are always equal to 27, and they are either division Jordan algebras (in the sense of Jacobson) or reduced. Moreover, if an Albert algebra \(J\) over \(\mathbb{K}\) is reduced, then it is isomorphic to the Jordan algebra of all \(3 \times 3\) matrices with entries in an octonion algebra over \(\mathbb{K}\) (uniquely determined by \(J\) and called the coordinate algebra of \(J\)) which are hermitian relative to a canonical involution. Therefore the problem of classifying Albert algebras centers in those that are division algebras. Since, up to a suitable scalar extension if necessary, every Albert algebra over \(\mathbb{K}\) becomes reduced (over the extended field), a tool to attack the above problem is to consider, for a given Albert algebra \(J\) over \(\mathbb{K}\), a ``reducing field'' of \(J\), namely a field extension \(\mathbb{F}/\mathbb{K}\) such that \(J \otimes_\mathbb{K} \mathbb{F}\) is reduced. In the present paper the authors prove that, if \(J\) is an Albert algebra over \(\mathbb{K}\), then there exists a unique octonion algebra \(C\) over \(\mathbb{K}\) such that, for every reducing field \(\mathbb{F}\) of \(J\), \(C \otimes_\mathbb{K} \mathbb{F}\) is the coordinate algebra of \(J \otimes_\mathbb{K} \mathbb{F}\). This nice result, showing that division Albert algebras also have a ``coordinate octonion algebra'', improves and simplifies the proof of closely related facts previously obtained by \textit{J.-P. Serre} (private communication to the authors) and \textit{M. Rost} [``A descent property for Pfister forms'', (Preprint)]. The main result in the paper (Theorem 4.2) ``almost'' describes the ``coordinate octonion algebra'' \(C\) of an arbitrary Albert algebra \(J\) depending on the constituents used to build up \(J\) by means of the generalized second Tits construction.
scientific article; zbMATH DE number 783903