= Ordered Fields =

'''Ordered fields''' are [[Analysis/Fields|fields]] with an '''ordering'''.

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== Description ==

A [[Analysis/Sets|set]] does not inherently have an ordering, which would make it an [[Analysis/OrderedSets|ordered set]]. Nor does it inherently support addition and multiplication, which would make it a [[Analysis/Fields|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'''.

[[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.



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