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Prime Ideals of a Galois Number Field and its Subfields

In: The Theory of Algebraic Number Fields

Author

Listed:
  • David Hilbert

Abstract

A number field K which coincides with all its conjugate fields is called a Galois number field. If k is an arbitrary number field of degree m and k′,..., k (m−1) are the fields conjugate to k then a new field K can be composed from all the numbers of the fields k, k ′, ..., k (m−1); this field K is then a Galois number field which includes all the fields k, k ′,..., k (m−1) as subfields. Thus any arbitrary field k can always be thought of as a subfield of a Galois number field. It follows from this observation that in our investigation of the properties of algebraic numbers it will be no essential restriction to start with a Galois number field and then show how the factorisation laws for the ideals of such a field carry over to an arbitrary subfield.

Suggested Citation

  • David Hilbert, 1998. "Prime Ideals of a Galois Number Field and its Subfields," Springer Books, in: The Theory of Algebraic Number Fields, chapter 10, pages 79-88, Springer.
    • Handle: RePEc:spr:sprchp:978-3-662-03545-0_10
    DOI: 10.1007/978-3-662-03545-0_10
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