Subgroups of free profinite groups and large subfields of {Mathematical expression}

We prove that many subgroups of free profinite groups are free, and use this to give new examples of pseudo-algebraically closed subfields of {Mathematical expression} satisfying Hilbert's Irreducibility Theorem, and to solve problems posed by M. Jarden and A. Macintyre. We also find a subfield of {Mathematical expression} which does not satisfy Hilbert's Irreducibility Theorem, but all of whose proper finite extensions do.