\(2\)-Galois groups and the Kaplansky radical

Abstract

An accurate description of the Galois group \(G_{F}(2)\) of the maximal Galois \(2\)-extension of a field \(F\) may be given for fields \(F\) admitting a \(2\)-henselian valuation ring. In this note we generalize this result by characterizing the fields for which \({G_{F}{(2)}}\) decomposes as a free pro-\(2\) product \(\mathcal{F}*\mathcal{H}\) where \(\mathcal{F}\) is a free closed subgroup of \({G_{F}{(2)}}\) and \(\mathcal{H}\) is the Galois group of a \(2\)-henselian extension of \(F\). The free product decomposition of \({G_{F}{(2)}}\) is equivalent to the existence of a valuation ring compatible with the Kaplansky radical of \(F\). Fields with Kaplansky radical fulfilling prescribed conditions are constructed, as an application.