© 2002 by London Mathematical Society
© The London Mathematical Society
On the Classification of 4-Dimensional Quadratic Division Algebras Over Square-Ordered Fields
Matematiska Institutionen, Uppsala Universitet Box 480, SE-751 06 Uppsala, Sweden, ernst.dieterich{at}math.uu.se
Matematiska Institutionen, Stockholms Universitet SE-106 91 Stockholm, Sweden, oehman{at}matematik.su.se
Received 2 March 2001. Revision received 26 September 2001.
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 x 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 pair problem defined by requiring Y+Yt to be orthogonally diagonalizable. The classifying list is given in terms of a 9-parameter family of configurations in k3, 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 
of its purely imaginary hyperplane. Calling A diagonalizable in case 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 x 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.