Dirichlet’s theorem for polynomial rings

Abstract

We prove the following form of Dirichlet's theorem for polynomial rings in one indeterminate over a pseudo algebraically closed field F. For all relatively prime polynomials a(X), b(X) E F[X] and for every sufficiently large integer n there exist infinitely many polynomials c(X) E F[X] such that a(X) + b(X)c(X) is irreducible of degree n, provided that F has a separable extension of degree n.