Whereas Iwasawa's theory of p-cyclotomic extensions was inspired by Weil's theory of the characteristic polynomial of the Frobenius endomorphism of a function field over a finite field of constants, the authors of the present paper in turn take Iwasawa's theory as a sample for an analogous theory in the setting of function fields resp. curves over finite fields. The paper arose as a byproduct of the famous Mazur- Wiles proof of a conjecture (''Hauptvermutung'') of Iwasawa [the authors, Class fields of Abelian extensions of \({\mathbb{Q}}\), Invent. Math., 179-330 (1984)]. This conjecture states that, for a 1-dimensional, p-adic valued odd character \(\chi\) of the absolute Galois group \(G({\bar {\mathbb{Q}}}/{\mathbb{Q}})\) of the field \({\mathbb{Q}}\) of rational numbers having a conductor not divisible by \(p^ 2\), the zeroes of the Iwasawa characteristic polynomial are given, after a change of variables, by the zeroes of the Kubota-Leopoldt p-adic L-function \(L_ p(\omega \chi^{- 1},s)\) in the extended s-disc \(| s|_ p