  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 $\F_q[t]$.
