Irreducibility in generalized power series

TL;DR

This paper explores the existence of irreducible elements in generalized power series fields, extending known results to larger ordinal types beyond previously established bounds.

Contribution

It significantly enlarges the class of ordinals for which irreducibles exist in fields of generalized power series, surpassing earlier known limits.

Findings

Irreducibles exist for a broader range of ordinal types.

Extended the known bounds from  to larger ordinals.

Provides new insights into the structure of generalized power series fields.

Abstract

A classical tool in the study of real closed fields are the fields $K((G))$ of generalized power series (i.e., formal sums with well-ordered support) with coefficients in a field $K$ of characteristic 0 and exponents in an ordered abelian group $G$. In this paper we enlarge the family of ordinals $\alpha$ of non-additively principal Cantor degree for which $K((\mathbb{R}^{\le 0}))$ admits irreducibles of order type $\alpha$ far beyond $\alpha=\omega^2 $ and $\alpha = \omega^3$ known prior to this work.