Ordered field

An ordered filed is a field together with a total ordering of its elements that is compatible with the field operations.

An ordered field necessarily has characteristic 0 since the elements 0<1<1+1<1+1+1<\dots necessarily are all distinct$. Thus, an ordered field necessarily contains an infinite number of elements: a finite field cannot be ordered.

Every ordered field contains an ordered subfield that is isomorphic to rational numbers. Any Dedekind complete ordered field is isomorphic to the real numbers. Squares are necessarily non-negative in an ordered filed.

Definition A filed (F,+,\times) together with a total order \leq on F is an ordered field if the order satisfies the following properties for all a,b and $c$ in F:

  • if a\leq b then a+c\leq b+c, and
  • if 0\leq a and 0\leq b then 0\leq a\times b.
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