The space of orderings \(X_F\) of a formally real field \(F\) can be decomposed into Marshall’s equivalent classes. Two distinct orderings are in the same class if they are members of a 4-fan. In one of his previous papers [Commun. Algebra 21, No. 12, 4495–4511 (1993; Zbl 0799.12001)] the author associated a closure \(\hat{F}\) to any class \(C\) and proved that for some Pythagorean fields, especially for those with finitely many orderings, the partition of \(X_F\) into equivalence classes corresponds to a decomposition of \(G_F(2)\) as a free product in the category of pro-2 groups.
In the paper under review he generalizes Marshall’s equivalence relation from orderings to the case of arbitrary subgroups of the multiplicative group \(F^*\) of index \(p,\) where \(F\) is a field of characteristic \(\neq p\) and containing the \(p\)th roots of unity.
The main results are generalizations of the results of the paper mentioned above, i.e. (1) if \(\hat{F}\) is a closure at a \(p\)-equivalence class \(C\), then \(G_{\hat{F}}(p)\) cannot be written nontrivially as a free pro-\(p\) product of two closed subgroups, (2) if \(G_F(p)=G_{L_1}(p)\star_p ...\star_p G_{L_n}(p)\) for some intermediate fields \(F\subset L_1,...,L_n\subset F(p)\), then each free factor \(G_{Li}(p)\) comes from \(G_{\hat{F}}(p)\) for the closure \(\hat{F}\) at some \(p\)-equivalence class, (3) if \(\hat{F}\) is the closure at a finite \(p\)-equivalence class \(C\), then \(C\) consists of all restrictions \(F\cap \hat{T}\) to \(F\) of subgroups \(\hat{T}\) of \(\hat{F}^*\) of index \(p.\)
The final section contains a discussion of the so called elementary decomposition of \(G_F(p)\), which is a counterpart of the elementary decomposition of the finitely generated Witt ring in the theory of quadratic forms.