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Math Help - Ordered Field definition/description

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    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
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    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.
    Last edited by Deveno; April 21st 2011 at 01:03 PM.
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