On the normalizer of finitely generated subgroups of absolute galois groups of uncountable hilbertian fields of characteristic 0 | Zendy

Wulf-Dieter Geyer | Zendy; Moshe Jarden | Zendy

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On the normalizer of finitely generated subgroups of absolute galois groups of uncountable hilbertian fields of characteristic 0

Author(s) -

Wulf-Dieter Geyer,

Moshe Jarden

Publication year - 1988

Publication title -

israel journal of mathematics

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 1.168

H-Index - 63

eISSN - 1565-8511

pISSN - 0021-2172

DOI - 10.1007/bf02778037

Subject(s) - mathematics , uncountable set , centralizer and normalizer , countable set , cardinality (data modeling) , algebraically closed field , combinatorics , complement (music) , discrete mathematics , pure mathematics , biochemistry , chemistry , complementation , computer science , gene , data mining , phenotype

For a fieldK and a positive integere let N e (K) be the set of alle-tuplesσ = (σ 1, …,σ e)εG(K) e that generate a selfnormalizer closed subgroup ofG(K). Chatzidakis proved, that ifK is Hilbertian and countable, then N e (K) has Haar measure 1. IfK is Hilbertian and uncountable, this need not be the case. Indeed, we prove that ifK 0 is a field of characteristic 0 that contains all roots of unity,T is a set of cardinality ℵ1 which is algebraically independent overK 0 andK =K 0(T), then neither N e (K) nor its complement contain a set of positive measure. In particular N e (K) is a nonmeasurable set.

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