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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 Hahn series ： ウィキペディア英語版 Hahn series In mathematics, Hahn series (sometimes also known as Hahn-Mal'cev-Neumann series) are a type of formal infinite series. They are a generalization of Puiseux series (themselves a generalization of formal power series) and were first introduced by Hans Hahn in 1907〔Hahn (1907)〕 (and then further generalized by Anatoly Maltsev and Bernhard Neumann to a non-commutative setting). They allow for arbitrary exponents of the indeterminate so long as the set supporting them forms a well-ordered subset of the value group (typically $\backslash mathbb$ or $\backslash mathbb$). Hahn series were first introduced, as groups, in the course of the proof of the Hahn embedding theorem and then studied by him as fields in his approach to Hilbert's seventeenth problem. ==Formulation== The field of Hahn series $KT^\backslash Gamma$ (in the indeterminate ''T'') over a field ''K'' and with value group ''Γ'' (an ordered group) is the set of formal expressions of the form $f\; =\; \backslash sum\_\; c\_e\; T^e$ with $c\_e\; \backslash in\; K$ such that the support $\backslash$ of ''f'' is well-ordered. The sum and product of $f\; =\; \backslash sum\_\; c\_e\; T^e$ and $g\; =\; \backslash sum\_\; d\_e\; T^e$ are given by $f+g\; =\; \backslash sum\_\; (c\_e+d\_e)\; T^e$ and $fg\; =\; \backslash sum\_\; \backslash sum\_\; c\_\; d\_\; T^e$ (in the latter, the sum $\backslash sum\_\; \backslash cdot$ over values $(e\text{'},e\text{'}\text{'})$ such that $c\_\; \backslash neq\; 0$ and $d\_\; \backslash neq\; 0$ is finite because a well-ordered set cannot contain an infinite decreasing sequence). For example, $T^\; +\; T^\; +\; T^\; +\; \backslash cdots$ is a Hahn series (over any field) because the set of rationals $\backslash ,\; -\backslash frac,\; -\backslash frac,\; \backslash ldots\backslash \}$ is well-ordered; it is not a Puiseux series because the denominators in the exponents are unbounded. (And if the base field ''K'' has characteristic ''p'', then this Hahn series satisfies the equation $X^p-X\; =\; T^$ so it is algebraic over $K(T)$.) The valuation $v(f)$ of $f\; =\; \backslash sum\_\; c\_e\; T^e$ is defined as the smallest ''e'' such that $c\_e\; \backslash neq\; 0$ (in other words, the smallest element of the support of ''f''): this makes $KT^\backslash Gamma$ into a spherically complete valued field with value group ''Γ'' (justifying ''a posteriori'' the terminology); in particular, ''v'' defines a topology on $KT^\backslash Gamma$. If $\backslash Gamma\; \backslash subseteq\; \backslash mathbb$, then ''v'' corresponds to an ultrametric) absolute value $|f|\; =\; \backslash exp(-v(f))$, with respect to which $KT^\backslash Gamma$ is a complete metric space. However, unlike in the case of formal Laurent series or Puiseux series, the formal sums used in defining the elements of the field do ''not'' converge: in the case of $T^\; +\; T^\; +\; T^\; +\; \backslash cdots$ for example, the absolute values of the terms tend to 1 (because their valuations tend to 0), so the series is not convergent (such series are sometimes known as "pseudo-convergent"〔Kaplansky (1942, ''Duke Math. J.'', definition on p.303)〕). If ''K'' is algebraically closed (but not necessarily of characteristic zero) and ''Γ'' is divisible, then $KT^\backslash Gamma$ is algebraically closed.〔MacLane (1939, ''Bull. Amer. Math. Soc.'', theorem 1 (p.889))〕 Thus, the algebraic closure of $K((T))$ is contained in $KT^$ has cardinality (strictly) less than ''κ'': it turns out that this is also a field, with much the same algebraic closedness properties as the full $KT^\backslash Gamma$: e.g., it is algebraically closed or real closed when ''K'' is so and ''Γ'' is divisible.〔Alling (1987, §6.23, (3) and (4) (p.218–219))〕 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「Hahn series」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.