On Crossed Homomorphisms and the Construction of a New Class of Locally Compact Dickson Nearfields

The aim of this paper is the construction of a new class of locally compact nearfields. They are all derived from a local field F by means of certain mappings $F^{\ast }\to \Delta \le AutF$, called couplings on F. Couplings on any field F can be described in terms of crossed homomorphisms $\Delta \to F^{\ast }/U$, where $U\le F^{\ast }$ is invariant under $\Delta$ and $[F^{\ast }:U]=|\Delta |$.