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Dedekind Theory of Ideals

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

Abstract

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).

Keywords

Prime Ideal Ideal Theory Number Field Integral Ideal Algebraic Integer 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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