Dedekind Theory of Ideals

Abstract

In Chapter I we studied the ring of integers o (p) of a single non-archimedean spot p. We shall see in § 33 J that the set of algebraic integers of a number field F can be expressed in the form

$$ o(S) = \mathop \cap \limits_{P \in S} o(p) $$

where S consists of all non-archimedean spots on F. This exhibits a strong connection between the algebraic integers and the prime spots of a number field, and we shall start to exploit it here. Specifically, we shall use the theory of prime spots to set up an ideal theory in o (S). For the present we can be quite general and we consider an arbitrary field F that is provided with a set of spots satisfying certain axioms. We shall call these axioms the Dedekind axioms for S since they lead to Dedekind’s ideal theory in o (S).