ON THE CLASSIFICATION OF $4$-DIMENSIONAL QUADRATIC DIVISION ALGEBRAS OVER SQUARE-ORDERED FIELDS

Abstract

A square-ordered field, also called a Hilbert field of type (A), is understood to be an ordered field all of whose positive elements are squares. The problem of classifying, up to isomorphism, all $4$ -dimensional quadratic division algebras over a square-ordered field $k$ is shown to be equivalent to the problem of finding normal forms for all pairs $(X, Y)$ of $3 \times 3$ matrices over $k, X$ being antisymmetric and $Y$ being positive definite, under simultaneous conjugation by ${\rm SO}_3(k)$ . A solution is derived for the subproblem of this matrix pair problem defined by requiring $Y+Y^t$ to be orthogonally diagonalizable. The classifying list is given in terms of a $9$ -parameter family of configurations in $k^3$ , formed by a pair of points and an ellipsoid in normal position.

Each $4$ -dimensional quadratic division algebra $A$ over a square-ordered field $k$ is shown to determine, uniquely up to sign, a self-adjoint linear endomorphism $\alpha$ of its purely imaginary hyperplane. Calling A diagonalizable in case $\alpha$ is orthogonally diagonalizable, the achieved solution of the matrix pair subproblem yields a full classification of all diagonalizable $4$ -dimensional quadratic division $k$ -algebras. This generalizes earlier results of both Hefendehl-Hebeker who classified, over Hilbert fields, those $4$ -dimensional quadratic division algebras having infinite automorphism group, and Dieterich, who achieved a full classification of all real $4$ -dimensional quadratic division algebras.

Finally, the paper describes explicitly how Hefendehl-Hebeker's classifying list, given in terms of a $4$ -parameter family of pairs of definite $3 \times 3$ matrices over $k$ , embeds into the classifying list of configurations. The image of this embedding turns out to coincide with the sublist of the list formed by all non-generic configurations.