A Class of Locally Compact Near-Fields Constructed from $\mathfrak {p}$-Adic Division Algebras

Abstract

We describe the construction of a specific class of disconnected locally compact near-fields. They are so-called Dickson near-fields and derived from $\mathfrak {p}$-adic division algebras by means of a special kind of homomorphisms or antihomomorphisms from the multiplicative group into the group of inner automorphisms of the division algebra. So let F be a local field and D be a finite-dimensional central division algebra over F. We presuppose that D/F is tamely ramified. In the first part of this paper we determine all finite subgroups of $D^*/F^*$. Based on that, we then determine all homomorphic and antihomomorphic couplings $D^*\rightarrow {{\,\textrm{Inn}\,}}(D)=D^*/F^*$ with finite image. With each of these couplings a locally compact near-field can be constructed from D. Apart from isomorphism, there is only a finite number of them. Compared to a previous publication, we omit the assumption that the image of the couplings is an Abelian group.