Minimal pairs, minimal fields and implicit constant fields

Abstract

Minimal pairs of definition were introduced by Alexandru, Popescu and Zaharescu [3,4,5] to study residue transcendental extensions. In this paper we obtain analogous results in the value transcendental case. We introduce the notion of minimal fields of definition for valuation transcendental extensions and show that they share some common ramification theoretic properties. The connection between minimal fields of definition and implicit constant fields is also investigated. Further, we explore the relationship between valuation transcendental extensions and pseudo-Cauchy sequences.