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The bottom theorem. (English) Zbl 0565.12011
Let K be a Hilbertian field and G(K) its absolute Galois group. Let e be a positive integer. Then for almost all e-tuples \(\sigma \in G(K)^ e\) the fixed field of \(\sigma\) is an infinite extension of any proper subfield containing K.

MSC:
12F10 Separable extensions, Galois theory
12F99 Field extensions
12F20 Transcendental field extensions
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References:
[1] D.Haran, Non-free torsion free profinite groups with open free subgroups. To appear in Israel. J. Math. · Zbl 0589.20017
[2] M. Jarden, Algebraic extensions of finite corank of Hilbertian fields. Israel. J. Math.18, 279-307 (1974). · Zbl 0293.12101 · doi:10.1007/BF02757283
[3] M. Jarden, An analogue of Artin-Schreier theorem. Math. Ann.242, 193-200 (1979). · Zbl 0416.12012 · doi:10.1007/BF01420725
[4] L.Ribes, Introduction to profinite groups and Galois cohomolgy. Queen’s Papers in Pure and Appl. Math.24, Kingston 1970. · Zbl 0221.12013
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