Compositions of consistent systems of rank one discrete valuation rings

TL;DR

This paper studies the conditions under which a set of invariants related to extensions of rank one discrete valuation rings can be realized by a suitable finite separable field extension.

Contribution

It introduces the concept of m-consistent systems and investigates their realizability in the context of algebraic field extensions.

Findings

Characterization of realizable m-consistent systems

Conditions for the existence of extensions matching given invariants

Insights into the structure of integral closures in field extensions

Abstract

Let V be a rank one discrete valuation ring (DVR) on a field F and let L/F be a finite separable algebraic field extension with [L:F] = m. The integral closure of V in L is a Dedekind domain that encodes the following invariants: (i) the number of extensions of V to a valuation ring W on L, (ii) the residue degree of each W over V, and (iii) the ramification degree of each W over V. Given a finite set of DVRs on F, an m-consistent system is a family of sets enumerating what is theoretically possible for the above invariants of each V in the set. The m-consistent system is realizable if there exists a finite separable extension field L/F that gives for each V the listed invariants. We investigate the realizability of m-consistent systems.