Subgroups of free profinite groups and large subfields of\(\mathop Q\limits^ \sim \)
- 147 Downloads
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
Unable to display preview. Download preview PDF.
- 5.A. Douady,Cohomologie des groupes compacts totalement discontinus, Séminaire Bourbaki, 1959–1960, exposé 189.Google Scholar
- 6.L. van den Dries,Decidable PAC-fields of algebraic numbers, in preparation.Google Scholar
- 13.M. Jarden,An analogue of Cebotarev density theorem for fields of finite corank, preprint.Google Scholar
- 15.A. Lubotzky,On the non-congruence structure of SL2, in preparation.Google Scholar
- 16.A. Lubotzky,Combinatorial group theory for pro-p-groups, in preparation.Google Scholar
- 19.L. Ribes,Introduction of profinite groups and Galois cohomology, Queens papers in pure and applied mathematics, no. 24 (1970).Google Scholar