  We prove an Iwasawa Main Conjecture for the class group of the
$\mathfrak{p}$-cyclotomic extension $\mathcal{F}$ of the function field
$\mathbb{F}_q(\theta)$ ($\mathfrak{p}$ is a prime of $\mathbb{F}_q[\theta]\,$),
showing that its Fitting ideal is generated by a Stickelberger element. We use
this and a link between the Stickelberger element and a $\mathfrak{p}$-adic
$L$-function to prove a close analog of the Ferrero-Washington theorem for
$\mathcal{F}$ and to provide informations on the $\mathfrak{p}$-adic valuations
of the Bernoulli-Goss numbers $\beta(j)$ (i.e., on the values of the Goss
$\zeta$-function at negative integers).
