A class of locally compact near-fields constructed from \(\mathfrak{p} \)-adic division algebras (Q6136254)

The author describes the construction of a certain class of disconnected locally compact near-fields. They are so-called Dickson near-fields and are derived from \(p\)-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. Dickson near-fields have a very special place in the history of algebra. In his study on the independence of the field axioms, Dickson (1905) developed a method using, what is now called, a coupling map for adjusting the multiplication of a finite field to produce a skew-field which satisfies only one distributive law. Such algebraic structures are called near-fields and the near-fields arising from the Dickson method are called Dickson near-fields. Near-fields are not just an algebraic curiosity. Amongst many other applications, Zassenhaus used them to describe the strongly transitive permutation groups and he also showed that, apart from seven strays, all finite near-fields are such Dickson near-fields. Moreover, Veblen and Wedderburn used near-fields to give examples of nondesarguesian planes. The locally compact connected near-fields have been classified by \textit{F. Kalscheuer} [Abh. Math. Semin. Univ. Hamb. 13, 413--435 (1940; Zbl 0023.00602)] and these results were extended by \textit{T. Grundhöfer} [Forum Math. 1, No. 1, 81--101 (1989; Zbl 0649.20045)] who showed that every locally compact near-field of characteristic \(0\) is a Dickson near-field derived from a locally compact skew-field. It is known that each locally compact non-connected skew field is a local field or a (proper) finite-dimensional division algebra over a local field, also called a \(p\)-adic division algebra. Thus, for a description of all locally compact near-fields, knowing the couplings on \(p\) -adic division algebras are important. The work presented here is a more general continuation of the author's earlier work in [\textit{D. Gröger}, Result. Math. 75, No. 4, Paper No. 170, 9 p. (2020; Zbl 1480.12008)] and relies on determining all suitable homomorphic and antihomomorphic coupling maps on certain groups.