On Archimedean Decompositions of Linearly Ordered Fields

TL;DR

This paper explores how complete ordered fields can be uniquely decomposed into a base field and multiple ordered Archimedean groups, extending Hahn's classical techniques with new invariants and functorial methods.

Contribution

It introduces a new invariant for ordered fields and refines Hahn's decomposition techniques to achieve a unique, multi-layered decomposition of complete ordered fields.

Findings

Complete ordered fields can be decomposed into a base field and ordered groups.

The decomposition into a field and groups is unique and can be extended to multiple groups.

New functorial methods and invariants facilitate this refined decomposition.

Abstract

In 1907, Hans Hahn proved the remarkable fact that any ordered group can be embedded in an ordered real function space. This set the stage for work on ordered groups and fields, and this area received valuable contributions from Levi-Civita and many others. In this paper we show how Hahn's method of generating ordered fields from a base field and an ordered group actually characterizes all complete ordered fields. We develop refinements of Hahn's techniques, show that they are functorial in nature, and develop a new invariant of an ordered field to prove that the decomposition of a complete field into a base field and a "group of exponents" can itself further decompose, uniquely, into a field combined with an arbitrary number of groups, all of which are ordered and Archimedean.