Abstract
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.
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The first author was supported by NSF grant MCS76-11625.
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Lubotzky, A., van den Dries, L. Subgroups of free profinite groups and large subfields of\(\mathop Q\limits^ \sim \) . Israel J. Math. 39, 25–45 (1981). https://doi.org/10.1007/BF02762851
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DOI: https://doi.org/10.1007/BF02762851