Authenticate Ordered Field For Free

Properties of ordered fields Either ?a ? 0 ? a or a ? 0 ? ?a. One can "add inequalities": if a ? b and c ? d, then a + c ? b + d. One can "multiply inequalities with positive elements": if a ? b and 0 ? c, then ac ? bc. Transitivity of inequality: if a < b and b < c, then a < c. If a < b and a, b > 0, then 1/b < 1/a.

The answer must be (b) the rational numbers. The rational numbers are a sub-field of the real numbers, and inherit the order of the latter, most basic example of an ordered field.

Multiplying both sides by ?i again, we get that 0 < (?1) · (?i) = i. Hence, if i < 0, then then 0 > i, again contradicting the second condition in the definition of an order. Hence, since we have that neither 0 < i nor i < 0, we see that there cannot exist an order on C that makes C into an ordered field.

Definition of an ordered field: An ordered field is a field containing a subset of elements closed under addition and multiplication and having the property that every element in the field is either 0, in the subset, or has its additive inverse in the subset.

A set F together with two operations + and ? and a relation < satisfying the 13 axioms above is called an ordered field. Thus the real numbers are an example of an ordered field. Another example of an ordered field is the set of rational numbers Q with the familiar operations and order.

Ordered set, we can have x ~ y . An "ordered field" however, is usually taken to be a totally ordered field -- so without extra conditions, x~y => x=y . "Partially ordered fields" have been studied, but you have to specify what conditions you mean.

Definition. A complete ordered field is an ordered field F with the least upper bound property (in other words, with the property that if S ? F, S = ? and S is bounded above then S has a least upper bound supS). Example 14. The real numbers are a complete ordered field.