For a prime number l and for a finite Galois l-extension of function fields L / K over an algebraically closed field of characteristic p <> l, it is obtained the Galois module structure of the generalized Jacobian associated to L, l and the ramified prime divisors. In the cyclic case an implicit integral representation of the Jacobian is obtained and this representation is compared with the explicit representation.

The main results of this paper may be loosely stated as follows.Theorem.— Let $N$ and $N^{\text{'}}$ be sums of Galois algebras with group $\Gamma$ over algebraic number fields. Suppose that $N$ and $N^{\text{'}}$ have the same dimension $\mathbb{Q}$ and that they are identical at their wildly ramified primes. Then (writing ${𝒪}_N$ for the maximal order in $N$)$${𝒪}_N\oplus {𝒪}_N\oplus \mathbb{Z}\Gamma {\cong }_{\mathbb{Z}\Gamma }\oplus {𝒪}_N^{\text{'}}\oplus {𝒪}_{N^{\text{'}}}\oplus \mathbb{Z}\Gamma .$$In many cases ${𝒪}_N{\cong }_{\mathbb{Z}\Gamma }{𝒪}_{N^{\text{'}}}.$The role played by the root numbers of $N$ and $N^{\text{'}}$ at the symplectic characters of $\Gamma$ in determining the relationship between the $\mathbb{Z}\Gamma$-modules ${𝒪}_N$ and ${𝒪}_{N^{\text{'}}}$ is described. The theorem includes...