Stable valued fields

TL;DR: In this paper, an extension of a notion in the monograph by S. Bosch, U. Guntzer, and R. Remmert (Non-Archimedean Analysis), namely, that of a (ultrametric) norm on groups, rings, algebras, and vector spaces, was proposed.

Abstract: We are concerned with a class of valued fields, called stable. We propound an extension of a notion in the monograph by S. Bosch, U. Guntzer, and R. Remmert (Non-Archimedean Analysis. A Systematic Approach to Rigid Analytic Geometry, Springer, Berlin (1984)), namely, that of a (ultrametric) norm on groups, rings, algebras, and vector spaces, to the case where the value of the norm is taken from an arbitrary (not necessarily Archimedean) linearly ordered Abelian group (using — as in the general theory of valuations — the version of a logarithmic norm). Our main result extends Proposition 6 in the cited monograph to the general case, thereby making it possible to use the technique of Cartesian spaces to deliver further results on stable valued fields.