The PAC Nullstellensatz. The
absolute Galois group of a field is
profinite, hence
compact, and hence equipped with a normalized
Haar measure. Let be a countable
Hilbertian field and let be a positive
integer. Then for almost all -tuples , the fixed field of the
subgroup generated by the
automorphisms is PAC. Here the phrase "almost all" means "all but a set of
measure zero".[5] (This result is a consequence of Hilbert's irreducibility theorem.)
Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Vol. 11 (3rd revised ed.).
Springer-Verlag.
ISBN978-3-540-77269-9.
Zbl1145.12001.
The PAC Nullstellensatz. The
absolute Galois group of a field is
profinite, hence
compact, and hence equipped with a normalized
Haar measure. Let be a countable
Hilbertian field and let be a positive
integer. Then for almost all -tuples , the fixed field of the
subgroup generated by the
automorphisms is PAC. Here the phrase "almost all" means "all but a set of
measure zero".[5] (This result is a consequence of Hilbert's irreducibility theorem.)
Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Vol. 11 (3rd revised ed.).
Springer-Verlag.
ISBN978-3-540-77269-9.
Zbl1145.12001.