Eine Klasse topologischer Fastkörperebenen
Let \(F=(F,+\), ) be a commutative field, \(\alpha\) : \(F\circ \to^{\sim}\Gamma =Aut F\), where \(x\circ y=x^{\alpha (y)}y\) for x,y\(\neq 0\). Then \(F_{\alpha}=(F,+,\circ)\) is a Dickson nearfield. Assume, moreover, that \(\alpha\) (F\(\circ)\) is finite and that w is a non- trivial valuation of F such that \(F_{\alpha}\) is a topological nearfield with respect to the topology \(\tau\) defined on F by w. Then \((F_{\alpha},\tau)\) coordinatizes a topological projective plane. Planarity of \(F_{\alpha}\) and continuity of the solution of \(x\circ a- x\circ b=c\) are fairly easy; the main part of the paper is devoted to the proof that join and intersection are continuous everywhere. If F is a local field of characteristic 0 and F:K is finite, then any map \(\alpha\) : \(F\circ \to^{\sim}Gal_ KF\) leads to a locally compact nearfield and hence to a compact 0-dimensional projective plane. The case of prime characteristic is more complicated, cp. also the author, Geom. Dedicata 20, 93-119 (1986). Both paper have been edited posthumousely by Th. Grundhöfer and H. Lüneburg.
scientific article; zbMATH DE number 3921375