Israel Journal of Mathematics

, Volume 39, Issue 1–2, pp 25–45 | Cite as

Subgroups of free profinite groups and large subfields of\(\mathop Q\limits^ \sim \)

  • A. Lubotzky
  • L. van den Dries


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.


Normal Subgroup Finite Index Open Subgroup Subnormal Subgroup Profinite Group 
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Copyright information

© The Weizmann Science Press of Israel 1981

Authors and Affiliations

  • A. Lubotzky
    • 1
    • 2
  • L. van den Dries
    • 1
    • 2
  1. 1.Department of MathematicsBar-Ilan UniversityRamat GanIsrael
  2. 2.Department of MathematicsYale UniversityNew HavenUSA

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