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that is compatible with the algebraic operations in the following sense:
It follows from these axioms that for every a, b, c, d in F:
Every subfield of an ordered field is also an ordered field. The smallest subfield is isomorphic to the rationals (as for any field of characteristic 0), and the order on this rational subfield is the same as the order of the rationals themselves. If every element of an ordered field lies between two elements of its rational subfield, then the field is said to be Archimedean. For example, the real numbers form an Archimedean field, but every hyperreal field is non-Archimedean.
If F is equipped with the order topology arising from the total order ≤, then the axioms guarantee that the operations + and * are continuous.
Examples of ordered fields are:
The surreal numbers form a proper classIn set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. Some classes are sets, for instance the class rather than a setThis article is about sets in mathematics. For other meanings, see Set (disambiguation). Sets are one of the most important and fundamental concepts in modern mathematics. Basic set theory, having only been invented at the end of the 19th century, is now, but otherwise obey the axioms of an ordered field. Every ordered field can be embedded into the surreal numbers.
Finite fieldIn abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theos cannot be turned into ordered fields, because they do not have characteristic 0. The complex numberThe complex numbers are an extension of the real numbers, in which all non-constant polynomials have roots. The complex numbers contain a number , the imaginary unit with , i. is a square root of. Every complex number can be represented in the form , whers also cannot be turned into an ordered field, as they contain a square rootIn mathematics, the square root of a non-negative real number is denoted and represents the non-negative real number whose square (the result of multiplying the number by itself) is. For example, since. This example suggests how square roots can arise whe of -1, which no ordered field can do. Also, the p-adic numbers cannot be ordered, since Q2 contains a square root of -7 and Qp (p > 2) contains a square root of 1-p. algebra