$\Theta$-Hilbertianity and strong $\Theta$-Hilbertianity
$\Theta$-Hilbertianity and strong $\Theta$-Hilbertianity
$\Theta$-Hilbertianity and its strengthening, strong $\Theta$-Hilbertianity, are two generalizations of Hilbertianity inspired by Jarden's definition of $p$-Hilbertianity and strong $p$-Hilbertianity. Jarden has asked whether the two notions defined by him are actually the same. We address this question in its more general version of $\Theta$-Hilbertianity and show that for PRC, and, in particular, for PAC fields, $p$-Hilbertianity and strong $p$-Hilbertianity coincide.
Comment: To appear in Israel Journal of Mathematics
PAC (pseudo algebraically closed) fields, Hilbertian fields, Mathematics - Number Theory, Separable extensions, Galois theory, Sylowian map, Hilbertian fields; Hilbert's irreducibility theorem, formally real field, PRC (pseudo real closed) fields, \(\Theta \)-Hilbertian fields, absolute Galois group, totally real extension totally real extension, Limits, profinite groups
PAC (pseudo algebraically closed) fields, Hilbertian fields, Mathematics - Number Theory, Separable extensions, Galois theory, Sylowian map, Hilbertian fields; Hilbert's irreducibility theorem, formally real field, PRC (pseudo real closed) fields, \(\Theta \)-Hilbertian fields, absolute Galois group, totally real extension totally real extension, Limits, profinite groups
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