# Thread: Ordered Field definition/description

1. ## Ordered Field definition/description

Hello,
What does it mean to be an ordered field? I can't find a definition in my Abstract Algebra book, nor in Wolfram.

Thanks,
Yvonne

2. 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.