Dirichlet's theorem for polynomial rings

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 $a(X), b(X)\in F[X]$ and for every sufficiently large integer $n$ there exist infinitely many polynomials $c(X)\in F[X]$ such that $a(X) + b(X)c(X)$ is irreducible of degree $n$, provided that $F$ has a separable extension of degree $n$.