Minimal pairs, minimal fields and implicit constant fields

TL;DR

This paper extends the concept of minimal pairs of definition to value transcendental extensions, introduces minimal fields of definition, and explores their relation to implicit constant fields and pseudo-Cauchy sequences.

Contribution

It introduces minimal fields of definition for valuation transcendental extensions and investigates their properties and connections to implicit constant fields.

Findings

Minimal fields of definition share ramification properties.

Established links between minimal fields and implicit constant fields.

Explored the relationship with pseudo-Cauchy sequences.

Abstract

Minimal pairs of definition were introduced by Alexandru, Popescu and Zaharescu 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.