On strong isotopy of Dickson semifields and geometric implications
An affine translation plane \({\mathcal P}\) is an incidence group. If \({\mathcal P}\) admits another abelian group acting regularly, then \({\mathcal P}\) can be coordinatized by a commutative semifield, and conversely. Moreover, \({\mathcal P}\) can be viewed as a commutative incidence group with the other group giving the group structure. Isomorphisms of these incidence groups can be characterized in terms of strong isotopisms of the corresponding multiplication variations of the underlying semifield. The authors study strong isotopisms of Dickson semifields. Special attention is payed to those constructed from absolutely algebraic fields and number fields. In particular, an example is given with countably many equivalence classes of strong isotopy.
scientific article; zbMATH DE number 1185383