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Orders in a Field

In: The Theory of Algebraic Number Fields

Author

Listed:
  • David Hilbert

Abstract

Let υ, η, ... be any algebraic integers whose domain of rationality is the field k l of degree m; then the set of all polynomials in ϑ, η, ... with rational integer coefficients is called an order 2. Addition, subtraction and multiplication of two numbers in an order produce again numbers in the order. An order is thus invariant under the three operations of addition, subtraction and multiplication. The maximal order in a field k is the order determined by ω 1, ... ω m where these are numbers of a basis for k; this consists of all the algebraic integers of k.

Suggested Citation

  • David Hilbert, 1998. "Orders in a Field," Springer Books, in: The Theory of Algebraic Number Fields, chapter 9, pages 67-75, Springer.
    • Handle: RePEc:spr:sprchp:978-3-662-03545-0_9
    DOI: 10.1007/978-3-662-03545-0_9
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