On the normalizer of finitely generated subgroups of absolute Galois groups of uncountable Hilbertian fields of characteristic 0
Main topic is the structure of absolute Galois groups of Hilbertian fields, i.e. fields where Hilbert's irreducibility theorem holds true. It is shown that if such fields contain only finitely many roots of unity, then for almost all (in the sense of Haar measure) automorphisms \(\delta_ 1,...,\delta_ e (e\geq 2)\) of the absolute Galois group G of K, the centralizer \(C_ G(<\delta >)\) in G of the subgroup generated by \(\delta_ 1,...,\delta_ e\) is trivial. On the contrary, if K contains all roots of unity, this is no longer true in general. For the Hilbertian rational field \(K=K_ 0(T)\) of characteristic zero with \(\aleph_ 1\) algebraically independent elements over \(K_ 0\), the subset of elements \(\delta_ 1,...,\delta_ e\) of \(G\times G\times...\times G\) such that \(C_ G(<\delta >)=1\) (or \(N_ G(<\delta >)=<\delta >)\) is not even measurable.
absolute Galois groups of Hilbertian fields
scientific article; zbMATH DE number 4091622