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Large normal extension of Hilbertian fields

Large normal extension of Hilbertian fields 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 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

Large normal extension of Hilbertian fields

Mathematische Zeitschrift , Volume 224 (4) – Apr 1, 1997

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References (13)

Publisher
Springer Journals
Copyright
Copyright © 1997 by Springer-Verlag Berlin Heidelberg
Subject
Legacy
ISSN
0025-5874
eISSN
1432-1823
DOI
10.1007/PL00004298
Publisher site
See Article on Publisher Site

Abstract

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

Journal

Mathematische ZeitschriftSpringer Journals

Published: Apr 1, 1997

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