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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 usethis 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 valuationsof the Bernoulli-Goss numbers $\beta(j)$ (i.e., on the values of the Goss$\zeta$-function at negative integers).

Iwasawa Main Conjecture for the Carlitz cyclotomic extension and applications