Growing Spines Ad Infinitum

TL;DR

This paper demonstrates that every non-trivial ordered abelian group can be extended with infinite elements and explores implications for definability of henselian valuations over fields of characteristic zero.

Contribution

It introduces the concept of augmentability of ordered abelian groups and connects it to the definability of henselian valuations in characteristic zero fields.

Findings

Every non-trivial ordered abelian group is augmentable by infinite elements.

A characterization of non-$t$-henselian fields in terms of definability of henselian valuations.

Establishes a link between group augmentability and valuation theory in field arithmetic.

Abstract

We show that every non-trivial ordered abelian group $G$ is augmentable by infinite elements, i.e., we have $G\preccurlyeq H\oplus G$ for some non-trivial ordered abelian group $H$. As an application, we show that when $k$ is a field of characteristic 0, then $k$ is not $t$-henselian if and only if all henselian valuations with residue field $k$ are ($\emptyset$-)definable.