Almost hilbertian fields

Almost Hilbertian Fields *

This paper is devoted to some variants of the Hilbert specialization property. For example, the RG-hilbertian property (for a field K), which arose in connection with the Inverse Galois Problem, requires that the specialization property holds solely for extensions of K(T ) that are Galois and regular over K. We show that fields inductively obtained from a real hilbertian field by adjoining real...

On Σ-hilbertian Fields

A field K is 0-Hilbertian if K 6= ⋃ni=1 φi(K) for any collection of rational functions φi of degree at least 2, i = 1, . . . ,m. Corvaja and Zannier [CoZ] give an elementary construction for a 0-Hilbertian field that isn’t Hilbertian. There is an obvious generalization of the notion of 0-Hilbertian to g-Hilbertian. Guralnick-Thompson and Liebeck-Saxl have given a partial classification of monod...

Galois Extensions of Hilbertian Fields

We prove the following result: Theorem. Let K be a countable Hilbertian field, S a finite set of local primes of K, and e ≥ 0 an integer. Then, for almost all ∈ G(K)e, the field Ks[ ] ∩Ktot,S is PSC. Here a local prime is an equivalent class p of absolute values of K whose completion is a local field, K̂p. Then Kp = Ks ∩ K̂p and Ktot,S = T p∈S T σ∈G(K) K σ p . G(K) stands for the absolute Galois ...

On the Characterization of Hilbertian Fields

The main goal of this work is to answer a question of Dèbes and Haran by relaxing the condition for Hilbertianity. Namely we prove that for a field K to be Hilbertian it suffices that K has the irreducible specialization property merely for absolutely irreducible polynomials.

Compositum of Galois Extensions of Hilbertian Fields