On ω-free PAC Fields

Abstract

The free profinite group ${\hat F_\omega }$ has this characterization: It is a profinite group of countable rank for which every finite embedding problem is solvable (Corollary 24.2). This property produces copies of ${\hat F_\omega }$ as closed subgroups of ${\hat F_e}$, for e≥2, in various ways (Section 24.3). In particular, if M is an open proper subgroup of a closed normal subgroup N of ${\hat F_e}$ of infinite index, then $M \cong {\hat F_\omega }$ (Proposition 24.7). Finally, with an additional hypothesis (Lemma 24.17) on the solutions of the finite embedding problem, many of the results of Sections 24.1 extend to groups of arbitrary rank (Section 24.4).