Ordered Field - Examples of Ordered Fields

Examples of Ordered Fields

Examples of ordered fields are:

  • the rational numbers
  • the real algebraic numbers
  • the computable numbers
  • the real numbers
  • the field of real rational functions, where p(x) and q(x), are polynomials with real coefficients, can be made into an ordered field where the polynomial p(x) = x is greater than any constant polynomial, by defining that whenever, for . This ordered field is not Archimedean.
  • The field of formal Laurent series with real coefficients, where x is taken to be infinitesimal and positive
  • real closed fields
  • superreal numbers
  • hyperreal numbers

The surreal numbers form a proper class rather than a set, but otherwise obey the axioms of an ordered field. Every ordered field can be embedded into the surreal numbers.

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