A topological characterization of norms on a nearfield (Q1193236)

The main purpose of this note is to extend to nearfields \(F\) the topological characterization of norms on (commutative) fields due to \textit{P. M. Cohn} [Proc. Camb. Philos. Soc. 50, 159-177 (1954; Zbl 0055.032)]. Thus the principal theorem states that a topology \(\mathcal T\) on \(F\) is defined by a nontrivial norm if, and only if, \((F,{\mathcal T})\) is a locally bounded semitopological nearring with non-zero topological nilpotent elements. Also given are extra restrictions on \(\mathcal T\) which determine whether or not the norm is non-Archimedean and/or stable. A norm \(p: F\to\mathbb{R}_ +\) gives rise to \(\mathcal T\) via the metric \((a,b)\to p(a-b)\) on \(F\); the precise description of the resulting topologies is somewhat technical. Starting from such a topology the author first constructs a gauge set of \(F\) in the manner of \textit{H.-J. Kowalsky} and \textit{H.-J. Dürbaum} [J. Reine Angew. Math. 191, 135-152 (1953; Zbl 0050.035)] in the case of fields from which he derives an associated norm.