Integrally Hilbertian rings and the polynomial Schinzel hypothesis

Abstract

We prove that all rings of integers of number fields are ``integrally Hilbertian''. That is, i.e. they satisfy an ``integral'' version of the classical Hilbert irreducibility property, to the effect that, under appropriate irreducibility assumptions on given multivariate polynomials with coefficients in such a ring ${\mathcal Z}$, one can specialize in the ring ${\mathcal Z}$ some of the variables in such a way that irreducibility over the ring ${\mathcal Z}$ is preserved for the specialized polynomials. We identify an intermediate type of ring, which we call ``Schinzel ring'', that is central in the study of this new Hilbert property. Dedekind domains and Unique Factorization Domains are shown to be Schinzel rings, leading to other examples of integrally Hilbertian rings. A main application is a polynomial version, for the ring ${\mathcal Z}[Y_1,\ldots,Y_n]$, of the Schinzel Hypothesis about primes in value sets of polynomials. Some consequences for the ring of integers itself are also deduced.

Theorems & Definitions (61)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Definition 2.1
• Definition 2.2
• Proposition 2.3
• Remark 2.4
• Definition 2.5
• Remark 2.6