ABSTRACT
This chapter focuses on extensions of rings of algebraic integers and of algebraic functions. It deals with integral extensions of Dedekind domains and shows how Dedekind domains can be characterized in terms of families of valuations satisfying the strong approximation theorem. The chapter describes two important invariants of an extension: the different and the discriminant. Divisibility in the ring of integers can be entirely described by valuations, and the same method clearly applies to any principal ideal domain, but in fact it holds for an even wider class of rings, of importance in what follows. The sets of valuations used will have to possess a property which is a strengthening of the approximation theorem; this can be shown to be equivalent to the uniqueness of ideal multiplication and lead to the class of Dedekind domains.
