a (linearly or totally) ordered field is a field F with an order "<" such that the trichotomy rule holds:
exactly one of the following is true for all a,b in F
a) a < b
b) b < a
c) a = b
furthermore, the order has to "respect addition and positivity":
a ≤ b implies a + c ≤ b + c, for all a,b,c in F (< is compatible with +)
0 ≤ a and 0 ≤ b implies 0 ≤ ab (< defines a set of positive elements closed under multiplication)
in particular, an ordered field must be of characteristic 0, since an order cannot be defined on a field of finite characteristic.
examples of ordered fields: Q, Q(√2), the field of algebraic real numbers, and R. among ordered fields, Q is minimal, any ordered field
contains a subfield isomorphic to the rationals.
some definitions instead define a set of "prepositive elements" P:
if x,y are in P, then x+y and xy are in P
for all x in F, x^2 is in P
-1 is not in P
if F = P U -P, then the elements in P - {0} are called "positive elements" and one defines the order by:
a < b iff b - a is in P. the advantage of this definition is that the ordering is defined in terms of the algebraic structure,
rather than considering a dual algebraic/preordered set from the beginning.