Ordered Field - Properties of Ordered Fields

Properties of Ordered Fields

• If x < y and y < z, then x < z. (transitivity)
• If x < y and z > 0, then xz < yz.
• If x < y and x,y > 0, then 1/y < 1/x

For every a, b, c, d in F:

• Either −a ≤ 0 ≤ a or a ≤ 0 ≤ −a.
• We are allowed to "add inequalities": If a ≤ b and c ≤ d, then a + c ≤ b + d
• We are allowed to "multiply inequalities with positive elements": If a ≤ b and 0 ≤ c, then ac ≤ bc.

• 1 is positive. (Proof: either 1 is positive or −1 is positive. If −1 is positive, then (−1)(−1) = 1 is positive, which is a contradiction)
• An ordered field has characteristic 0. (Since 1 > 0, then 1 + 1 > 0, and 1 + 1 + 1 > 0, etc. If the field had characteristic p > 0, then −1 would be the sum of p − 1 ones, but −1 is not positive). In particular, finite fields cannot be ordered.
• Squares are non-negative. 0 ≤ a² for all a in F. (Follows by a similar argument to 1 > 0)

Every subfield of an ordered field is also an ordered field (inheriting the induced ordering). The smallest subfield is isomorphic to the rationals (as for any other 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. Otherwise, such field is a non-Archimedean ordered field and contains infinitesimals. For example, the real numbers form an Archimedean field, but every hyperreal field is non-Archimedean.

An ordered field K is the real number field if it satisfies the axiom of Archimedes and every Cauchy sequence of K converges within K.