Primes Splitting in the Gaussian Integers - Function Field Analogue

Question

The function field analogue of the ring $\mathbb{Z}[i]$ are functions of the form $A(T)+\alpha B(T)$, where $\alpha$ is a solution to the equation $x^{2}+T=0$, over $F_{q}[T]$. We know that a prime $p$ splits over $\mathbb{Z}[i]$ if and only if $p \equiv 1$ mod $4$. Is there an analogous Dirichlet character that determines which primes split over $F_{q}[T](\alpha)$? More generally, how do I determine which primes split in a quadratic extension of $F_{q}[T]$?