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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 Stufe (algebra) ： ウィキペディア英語版 Stufe (algebra) In field theory, the Stufe (/ʃtuːfə/; German: level) ''s''(''F'') of a field ''F'' is the least number of squares that sum to -1. If -1 cannot be written as a sum of squares, ''s''(''F'')=$\backslash infty$. In this case, ''F'' is a formally real field. Albrecht Pfister proved that the Stufe, if finite, is always a power of 2, and that conversely every power of 2 occurs.〔 ==Powers of 2== If $s(F)\backslash ne\backslash infty$ then $s(F)=2^k$ for some $k\backslash in\backslash Bbb\; N$.〔Rajwade (1993) p.13〕〔Lam (2005) p.379〕 ''Proof:'' Let $k\; \backslash in\; \backslash Bbb\; N$ be chosen such that . Let $n\; =\; 2^k$. Then there are $s\; =\; s(F)$ elements $e\_1,\; \backslash ldots,\; e\_s\; \backslash in\; F\backslash setminus\backslash$ such that :$0\; =\; \backslash underbrace\_\; +\; \backslash underbrace\_\backslash ;.$ Both $a$ and $b$ are sums of $n$ squares, and $a\; \backslash ne\; 0$, since otherwise , contrary to the assumption on $k$. According to the theory of Pfister forms, the product $ab$ is itself a sum of $n$ squares, that is, $ab\; =\; c\_1^2\; +\; \backslash cdots\; +\; c\_n^2$ for some $c\_i\; \backslash in\; F$. But since $a+b=0$, we also have $-a^2\; =\; ab$, and hence :$-1\; =\; \backslash frac\; =\; \backslash left(\backslash frac\; \backslash right)^2\; +\; \backslash cdots\; +\backslash left(\backslash frac\; \backslash right)^2\backslash ;,$ and thus $s(F)\; =\; n\; =\; 2^k$. 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「Stufe (algebra)」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.