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Math. Z. 224, 555–565 (1997) c Springer-Verlag 1997 Large normal extension of Hilbertian fields Moshe Jarden School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv, 69978, Israel; e-mail: jarden@math.tau.ac.il Received 10 June 1994; in final form 15 May 1995 Introduction The goal of this note is to consider a certain natural family of closed normal subgroups of G (Q) and to prove that each group in this family is free. More generally, consider a countable separably Hilbertian field K . Denote the absolute Galois group of K by G (K ). Then, for almost all 2 G (K ) the field K ()is PAC and e-free [FJ2, Thms. 16.13 and 16.18]. Here K is the separable closure of K and K ( ) is the fixed field of in K . Being PAC means that every nonvoid s s absolutely irreducible variety defined over K ( ) has a K ( )-rational point. We s s say that K ()is e-free if G (K ()) (i.e., the closed subgroup h ;::: ; i of s s 1 e G (K ) generated by ;::: ; ) is free on
Mathematische Zeitschrift – Springer Journals
Published: Apr 1, 1997
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