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[[Analysis/Fields#Finite_Fields|Finite fields]] cannot be ordered. In the ''F,,2,,'' case, see that ''0 < 1'' should imply that ''0 + c < 1 + c'', but if ''c'' is 1 then it does not hold. |
Ordered Fields
Ordered fields are fields with an ordering.
Contents
Description
A set does not inherently have an ordering, which would make it an ordered set. Nor does it inherently support addition and multiplication, which would make it a field. An ordered field does both.
Such a set has several derived properties:
if a < b then a + c < b + c
if 0 < a and 0 < b then 0 < ab
if 0 < a then -a < 0
If an element of an ordered field is greater than 0, then it is positive. If it is greater than or equal to 0, then it is non-negative. If it is less than 0, then it is negative. If it is less than or equal to 0, then it is non-positive.
Finite fields cannot be ordered. In the F2 case, see that 0 < 1 should imply that 0 + c < 1 + c, but if c is 1 then it does not hold.