Random Galois extensions of Hilbertian fields
Let be a Galois extension of a countable Hilbertian field . Although need not be Hilbertian, we prove that an abundance of large Galois subextensions of are.
Reducibility of lacunary polynomials. X
Reduction and specialization of polynomials
We show explicit forms of the Bertini-Noether reduction theorem and of the Hilbert irreducibility theorem. Our approach recasts in a polynomial context the geometric Grothendieck good reduction criterion and the congruence approach to HIT for covers of the line. A notion of “bad primes” of a polynomial P ∈ ℚ[T,Y] irreducible over ℚ̅ is introduced, which plays a central and unifying role. For such a polynomial P, we deduce a new bound for the least integer t₀ ≥ 0 such that P(t₀,Y) is irreducible...