Classification of finite W-groups

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.