A fundamental identity for extensions of Dedekind domains.

The well-known fundamental identity in number theory expresses the degree of an extension of global fields in terms of local information. In this article we modify the fundamental identity so that it holds for arbitrary Dedekind domains. We also give necessary and sufficient conditions for the ground Dedekind domain such that the fundamental identity holds true. In particular, a Dedekind domain is excellent if and only if it is Japanese.