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.
Read more about this topic: Ordered Field
Famous quotes containing the words examples of, examples, ordered and/or fields:
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)
“In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
—Michel de Montaigne (1533–1592)
“Today everything is different. I can’t even get decent food. Right after I got here I ordered some spaghetti with marinara sauce and I got egg noodles and catsup. I’m an average nobody, I get to live the rest of my life like a schnook.”
—Nicholas Pileggi, U.S. screenwriter, and Martin Scorsese. Henry Hill (Ray Liotta)
“Mantled in grey, the dusk steals slowly in,
Crossing the dead, dull fields with footsteps cold.”
—Philip Larkin (1922–1986)