On the Classification of Four-Dimensional Quadratic Division Algebras over Square-Ordered Fields

A square-ordered field, also called 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× 3-matrices over k, X being antisymmetric and Y being positive definite, under simultaneous conjugation by SO3(k). A solution is derived for the subproblem of this matrix 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, formed by a pair of points and an ellipsoid in normal position. Each 4-dimensional quadratic division algebra A over a squareordered field k is shown to determine, uniquely up to sign, a self-adjoint linear endomorphism α of its purely imaginary hyperplane. Calling A diagonalizable in case α 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 [9] who classified, over Hilbert fields, those 4-dimensional quadratic division algebras having infinite automorphism group, and Dieterich [3],[4] who achieved a full classification of all real 4-dimensional quadratic division algebras. Finally we describe explicitly how Hefendehl-Hebeker’s classifying list, given in terms of a 4-parameter family of pairs of definite 3 × 3matrices over k, embeds into our classifying list of configurations. The image of this embedding turns out to coincide with the sublist of our list formed by all non-generic configurations.