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It is easy to see that these three properties are equivalent. It is also easy to see that a field that admits an ordering must satisfy these three properties.
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argument shows that the prepositive cone of sums of squares can be extended to a positive cone
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the element −1 is a sum of 1s.) This can be expressed in first-order logic by
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that can be equipped with a (not necessarily unique) ordering that makes it an
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is a field that also satisfies one of the following equivalent properties:
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in the language of fields and are equivalent to the above definition.
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and positive cones. Suppose −1 is not a sum of squares; then a
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equals zero, then each of those elements must be zero.
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is infinite. (In particular, such a field must have
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499:Milnor and Husemoller (1973) p.60
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551:Rajwade, A. R. (1993).
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110:December 2009
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532:(1973).
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