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An ordered field is a field F along with the prescription of a subset P (called the "positive half") of F such that the following conditions hold:
x,y in P, (x+y) and (xy) are both in P. (positive + positive = positive, positive * positive = positive)
x != 0 in F, either x is in P or -x is in P, but not both.

Here's why this makes sense. Once you've defined the "positive half" P of a field F, you can define what it means for one element x of F to be "greater than" a different element y (not equal to x), like so: If x-y is in the positive half P, then x is greater than y. If not, then by the second condition above, the opposite, y-x, must be in P, so y is greater than x. Thus there is some order to the elements of the field.
The rational numbers are an example of an ordered field, as are the real numbers and the integers.

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