Subgroups of free profinite groups and large subfields of\(\mathop Q\limits^ \sim \)
We prove that many subgroups of free profinite groups are free, and use this to give new examples of pseudo-algebraically closed subfields of\(\mathop Q\limits^ \sim \) satisfying Hilbert’s Irreducibility Theorem, and to solve problems posed by M. Jarden and A. Macintyre. We also find a subfield of\(\mathop Q\limits^ \sim \) which does not satisfy Hilbert’s Irreducibility Theorem, but all of whose proper finite extensions do.
KeywordsNormal Subgroup Finite Index Open Subgroup Subnormal Subgroup Profinite Group
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