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...

In this paper, under a mild hypothesis, we prove a conjecture of Gross for the Stickelberger element of the maximal abelian extension over the rational function field unramified outside a set of four degree-one places.