Classification of finite W-groups

TL;DR

This paper classifies the structure of W-groups associated with Pythagorean formally real fields when their space of orderings is finite, extending previous work and providing new Galois-theoretic proofs.

Contribution

It determines the structure of W-groups for finite spaces of orderings, generalizing Marshall's results and offering a Galois-theoretic proof of key lemmas.

Findings

W-groups are classified based on the connectedness of the space of orderings.

In the connected case, the structure reduces to that of the residue field.

In the disconnected case, the W-group is a free product of component W-groups.

Abstract

We determine the structure of the W-group $\mathcal{G}_F$, the small Galois quotient of the absolute Galois group $G_F$ of the Pythagorean formally real field $F$ when the space of orderings $X_F$ has finite order. Based on Marshall's work (1979), we reduce the structure of $\mathcal{G}_F$ to that of $\mathcal{G}_{\bar{F}}$, the W-group of the residue field $\bar{F}$ when $X_F$ is a connected space. In the disconnected case, the structure of $\mathcal{G}_F$ is the free product of the W-groups $\mathcal{G}_{F_i}$ corresponding to the connected components $X_i$ of $X_F$. We also give a completely Galois theoretic proof for Marshall's Basic Lemma.