  Recently the second author has associated a finite $\F_q[T]$-module $H$ to
the Carlitz module over a finite extension of $\F_q(T)$. This module is an
analogue of the ideal class group of a number field.
  In this paper we study the Galois module structure of this module $H$ for
`cyclotomic' extensions of $\F_q(T)$. We obtain function field analogues of
some classical results on cyclotomic number fields, such as the $p$-adic class
number formula, and a theorem of Mazur and Wiles about the Fitting ideal of
ideal class groups. We also relate the Galois module $H$ to Anderson's module
of circular units, and give a negative answer to Anderson's
Kummer-Vandiver-type conjecture.
  These results are based on a kind of equivariant class number formula which
refines the second author's class number formula for the Carlitz module.
