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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク P-adically closed field ： ウィキペディア英語版 P-adically closed field In mathematics, a ''p''-adically closed field is a field that enjoys a closure property that is a close analogue for ''p''-adic fields to what real closure is to the real field. They were introduced by James Ax and Simon B. Kochen in 1965.〔Ax & Kochen (1965)〕 == Definition == Let ''K'' be the field ℚ of rational numbers and ''v'' be its usual ''p''-adic valuation (with $v(p)=1$). If ''F'' is a (not necessarily algebraic) extension field of ''K'', itself equipped with a valuation ''w'', we say that $(F,w)$ is formally ''p''-adic when the following conditions are satisfied: * ''w'' extends ''v'' (that is, $w(x)=v(x)$ for all ''x'' in ''K''), * the residue field of ''w'' coincides with the residue field of ''v'' (the residue field being the quotient of the valuation ring $\backslash$ by its maximal ideal $\backslash$), * the smallest positive value of ''w'' coincides with the smallest positive value of ''v'' (namely 1, since ''v'' was assumed to be normalized): in other words, a uniformizer for ''K'' remains a uniformizer for ''F''. (Note that the value group of ''K'' may be larger than that of ''F'' since it may contain infinitely large elements over the latter.) The formally ''p''-adic fields can be viewed as an analogue of the formally real fields. For example, the field ℚ(i) of Gaussian rationals, if equipped with the valuation w given by $w(2+i)=1$ (and $w(2-i)=0$) is formally 5-adic (the place ''v''=5 of the rationals splits in two places of the Gaussian rationals since $X^2+1$ factors over the residue field with 5 elements, and ''w'' is one of these places). The field of 5-adic numbers (which contains both the rationals and the Gaussian rationals embedded as per the place ''w'') is also formally 5-adic. On the other hand, the field of Gaussian rationals is ''not'' formally 3-adic for any valuation, because the only valuation ''w'' on it which extends the 3-adic valuation is given by $w(3)=1$ and its residue field has 9 elements. When ''F'' is formally ''p''-adic but that there does not exist any proper ''algebraic'' formally ''p''-adic extension of ''F'', then ''F'' is said to be ''p''-adically closed. For example, the field of ''p''-adic numbers is ''p''-adically closed, and so is the algebraic closure of the rationals inside it (the field of ''p''-adic algebraic numbers). If ''F'' is ''p''-adically closed, then:〔Jarden & Roquette (1980), lemma 4.1〕 * there is a unique valuation ''w'' on ''F'' which makes ''F'' ''p''-adically closed (so it is legitimate to say that ''F'', rather than the pair $(F,w)$, is ''p''-adically closed), * ''F'' is Henselian with respect to this place (that is, its valuation ring is so), * the valuation ring of ''F'' is exactly the image of the Kochen operator (see below), * the value group of ''F'' is an extension by ℤ (the value group of ''K'') of a divisible group, with the lexicographical order. The first statement is an analogue of the fact that the order of a real-closed field is uniquely determined by the algebraic structure. The definitions given above can be copied to a more general context: if ''K'' is a field equipped with a valuation ''v'' such that * the residue field of ''K'' is finite (call ''q'' its cardinal and ''p'' its characteristic), * the value group of ''v'' admits a smallest positive element (call it 1, and say π is a uniformizer, i.e. $v(\backslash pi)=1$), * ''K'' has finite absolute ramification, i.e., $v(p)$ is finite (that is, a finite multiple of $v(\backslash pi)=1$), (these hypotheses are satisfied for the field of rationals, with ''q''=π=''p'' the prime number having valuation 1) then we can speak of formally ''v''-adic fields (or $\backslash mathfrak$-adic if $\backslash mathfrak$ is the ideal corresponding to ''v'') and ''v''-adically complete fields. 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「P-adically closed field」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.