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In mathematics, in particular in field theory and real algebra, a formally real field is a field that can be expanded with a (not necessarily unique) ordering that makes it an ordered field. ==Alternative Definitions== The definition given above is not a first-order definition, as it requires quantifiers over sets. However, the following criteria can be coded as first-order sentences in the language of fields, and are equivalent to the above definition. A formally real field ''F'' is a field that satisfies in addition one of the following equivalent properties:〔Rajwade, Theorem 15.1.〕〔Milnor and Husemoller (1973) p.60〕 * −1 is not a sum of squares in ''F''. In other words, the Stufe of ''F'' is infinite. (In particular, such a field must have characteristic 0, since in a field of characteristic ''p'' the element −1 is a sum of 1's.) * There exists an element of ''F'' that is not a sum of squares in ''F'', and the characteristic of ''F'' is not 2. * If any sum of squares of elements of ''F'' equals zero, then each of those elements must be zero. 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. A proof that if ''F'' satisfies these three properties, then ''F'' admits an ordering uses the notion of prepositive cones and positive cones. Suppose −1 is not a sum of squares, then a Zorn's Lemma argument shows that the prepositive cone of sums of squares can be extended to a positive cone ''P⊂F''. One uses this positive cone to define an ordering: ''a≤b'' if and only if ''b-a'' belongs to ''P''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「formally real field」の詳細全文を読む スポンサード リンク
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