Lokalkompakte Fastkörper. (Locally compact nearfields) (Q1104362)

Using Haar measures on the additive group of a nondiscrete locally compact field K, \textit{A. Weil} [``Basic Number Theory'', Grundlehren math. Wiss. 144 (1967; Zbl 0176.336)] introduced a valuation mod\(_ K: K\to {\mathbb{R}}.\) The present note does the same for disconnected, locally compact nearfields F and thus obtains a ``natural'' valuation \(| \quad |_ F.\) It turns out to be non-Archimedean, complete and discrete (Satz 3), and the valuation nearring R that arises has exactly one maximal ideal P, R/P being finite (Satz 4) while the subnearfield \(K_ F=\{k\in F: (x+y)k=xk+yk\) for all \(x,y\in F\}\) of F is closed (Satz 1g) and is either finite or non-discrete (Satz 1f). If \(K_ F\) is infinite and E is an infinite closed subfield of \(K_ F\), then F has finite E-dimension, and ramification of F/E can be introduced via the valuation.