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

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Results in Mathematics

Construction of Locally Compact Near-Fields from p-Adic Division Algebras Detlef Gr¨oger In memoriam Heinrich Wefelscheid

Abstract. The aim of this work is the construction of a new class of disconnected locally compact near-fields. They are all Dickson near-fields and derived from finite-dimensional division algebras over local fields by means of strong couplings with a finite Abelian Dickson group consisting of inner automorphisms. Mathematics Subject Classification. Primary 12K05; Secondary 11S45. Keywords. Locally compact near-fields, division algebras over local fields.

1. Introduction The interest in locally compact (nondiscrete) near-fields comes mainly from geometric investigations. For instance, these structures appear in introducing coordinates in certain compact projective planes; see [13] with further references. A particularly smooth class of near-fields consists of the so-called Dickson near-fields. Each of these structures is attached to a certain skew field, from which it can be derived by means of a mapping called coupling on the underlying skew field. This method was established in its full generality by Karzel [9]. The locally compact connected near-fields have been classified by Kalscheuer [8]: these are the fields of real and complex numbers and certain Dickson near-fields derived from the skew field of Hamilton’s quaternions. Grundh¨ ofer [4] extended this result and showed that every locally compact 0123456789().: V,-vol

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Results Math

near-field of characteristic 0 is a Dickson near-field derived from a locally compact skew field. It is well known that each locally compact non-connected skew field is a local field or a (proper) finite-dimensional division algebra over a local field [11, p. 173]. Sometimes these latter structures are also called p-adic division algebras. Thus, in order to determine all locally compact near-fields, one has to investigate the couplings on local fields and on p-adic division algebras. While there are a number of studies dedicated to the first problem [2,3,7,13,14], little is known about the second problem. In this note we consider a central division algebra D/F , which is tamely ramified above the local field F . Our goal is the determination of all strong (i.e. homomorphic) couplings κ on D, whose image is a finite and Abelian subgroup of Inn(D) = D∗ /F ∗ . These groups are determined completely in Theorem 3.6. Apart from conjugacy, there is only a finite number of them. Among these the Abelian subgroups of C/C ∩ F ∗ play an important role, where C is a fixed complement of the group of principal units in D∗ . By means of the reduced norm in D/F , the determination of the strong couplings κ can be transfered to the determination of all homomorphisms ψ : F ∗ → D∗ /F ∗ ; and if two such homomorphisms differ by an inner automorphism of D∗ /F ∗ , then the derived near-fields are isomorphic (Proposition 2.3). The task of describing these homomorphisms ψ relies essentially on the determination of the homomorphisms C ∩ F ∗ → C/C ∩ F ∗ ,