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In mathematics, a real closed field is a field ''F'' that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers. ==Definitions== A real closed field is a field ''F'' in which any of the following equivalent conditions are true: #''F'' is elementarily equivalent to the real numbers. In other words it has the same first-order properties as the reals: any sentence in the first-order language of fields is true in ''F'' if and only if it is true in the reals. (The choice of signature is not significant.) #There is a total order on ''F'' making it an ordered field such that, in this ordering, every positive element of ''F'' has a square root in ''F'' and any polynomial of odd degree with coefficients in ''F'' has at least one root in ''F''. #''F'' is a formally real field such that every polynomial of odd degree with coefficients in ''F'' has at least one root in ''F'', and for every element ''a'' of ''F'' there is ''b'' in ''F'' such that ''a'' = ''b''2 or ''a'' = −''b''2. #''F'' is not algebraically closed but its algebraic closure is a finite extension. #''F'' is not algebraically closed but the field extension is algebraically closed. #There is an ordering on ''F'' which does not extend to an ordering on any proper algebraic extension of ''F''. #''F'' is a formally real field such that no proper algebraic extension of ''F'' is formally real. (In other words, the field is maximal in an algebraic closure with respect to the property of being formally real.) #There is an ordering on ''F'' making it an ordered field such that, in this ordering, the intermediate value theorem holds for all polynomials over ''F'' with degree ''≥'' 0. #''F'' is a real closed ring. If ''F'' is an ordered field, the Artin–Schreier theorem states that ''F'' has an algebraic extension, called the real closure ''K'' of ''F'', such that ''K'' is a real closed field whose ordering is an extension of the given ordering on ''F'', and is unique up to a unique isomorphism of fields identical on ''F''〔Rajwade (1993) pp. 222–223〕 (note that every ring homomorphism between real closed fields automatically is order preserving, because ''x'' ≤ ''y'' if and only if ∃''z'' ''y'' = ''x'' + ''z''2). For example, the real closure of the ordered field of rational numbers is the field of real algebraic numbers. The theorem is named for Emil Artin and Otto Schreier, who proved it in 1926. If (''F'',''P'') is an ordered field, and ''E'' is a Galois extension of ''F'', then by Zorn's Lemma there is a maximal ordered field extension (''M'',''Q'') with ''M'' a subfield of ''E'' containing ''F'' and the order on ''M'' extending ''P''. ''M'', together with its ordering ''Q'', is called the relative real closure of (''F'',''P'') in ''E''. We call (''F'',''P'') real closed relative to ''E'' if ''M'' is just ''F''. When ''E'' is the algebraic closure of ''F'' the relative real closure of ''F'' in ''E'' is actually the real closure of ''F'' described earlier.〔Efrat (2006) p. 177〕 If ''F'' is a field (no ordering compatible with the field operations is assumed, nor is assumed that ''F'' is orderable) then ''F'' still has a real closure, which may not be a field anymore, but just a real closed ring. For example, the real closure of the field is the ring (the two copies correspond to the two orderings of ). On the other hand, if is considered as an ordered subfield of , its real closure is again the field . 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「Real closed field」の詳細全文を読む スポンサード リンク
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