2-Galois groups and the Kaplansky radical

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.