Irreducible values of polynomials

TL;DR

This paper proves an arithmetic analog of Schinzel's Hypothesis H for polynomial rings over pseudo algebraically closed fields, extending results to large finite fields using irreducibility theorems.

Contribution

It introduces a new proof of an analog of Schinzel's Hypothesis H in polynomial rings over certain fields, with implications for finite field results.

Findings

Proves an arithmetic analog of Schinzel's Hypothesis H.

Establishes irreducibility results akin to Hilbert's theorems.

Extends prime value results to large finite fields.

Abstract

Schinzel's Hypothesis H is a general conjecture in number theory on prime values of polynomials that generalizes, e.g., the twin prime conjecture and Dirichlet's theorem on primes in arithmetic progression. We prove an arithmetic analog of this conjecture for polynomial rings over pseudo algebraically closed fields. This implies results over large finite fields. A main tool in the proof is an irreducibility theorems \`a la Hilbert.