A Herbrand-Ribet theorem for function fields

We prove a function field analogue of the Herbrand-Ribet theorem on cyclotomic number fields. The Herbrand-Ribet theorem can be interpreted as a result about cohomology with ${\mu }_p$-coefficients over the splitting field of ${\mu }_p$, and in our analogue both occurrences of ${\mu }_p$ are replaced with the $\mathfrak{p}$-torsion scheme of the Carlitz module for a prime $\mathfrak{p}$ in ${\text{F}}_q[t]$.