Minimal pairs, inertia degrees, ramification degrees and implicit constant fields

TL;DR

This paper explores valuation transcendental extensions of valued fields, introducing an invariant linked to minimal pairs that reveals key information about the implicit constant field.

Contribution

It defines a new integer associated with valuation transcendental extensions, independent of minimal pair choice, and connects it to the implicit constant field.

Findings

The integer is uniquely determined by the extension.

It encodes significant information about the implicit constant field.

The integer is independent of the minimal pair chosen.

Abstract

An extension (K(X)|K, v) of valued fields is said to be valuation transcendental if we have equality in the Abhyankar inequality. Minimal pairs of definition are fundamental objects in the investigation of valuation transcendental extensions. In this article, we associate a uniquely determined positive integer with a valuation transcendental extension. This integer is defined via a chosen minimal pair of definition, but it is later shown to be independent of the choice. Further, we show that this integer encodes important information regarding the implicit constant field of the extension (K(X)|K, v).