Dedekind Theory of Ideals

  • O. Timothy O’Meara
Part of the Classics in Mathematics book series (volume 117)


In Chapter I we studied the ring of integers o (p) of a single non- archimedean spot p. We shall see in §33J 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).


Prime Ideal Ideal Theory Number Field Integral Ideal Algebraic Integer 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • O. Timothy O’Meara
    • 1
  1. 1.Department of MathematicsUniversity of Notre DameNotre DameUSA

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