  It is a classical result that prime numbers of the form $x^2+ny^2$ can be
characterized via class field theory for an infinite set of $n$. In this paper
we derive the function field analogue of the classical result. Then we apply an
effective version of the Chebotarev density theorem to bound the degree of the
smallest irreducible of the form $x^2-dy^2$, where $x$, $y$, and $d$ are
elements of a polynomial ring over a finite field.
