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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 Supernatural number ： ウィキペディア英語版 Supernatural number In mathematics, the supernatural numbers (sometimes called generalized natural numbers or Steinitz numbers) are a generalization of the natural numbers. They were used by Ernst Steinitz in 1910 as a part of his work on field theory. A supernatural number $\backslash omega$ is a formal product: : $\backslash omega\; =\; \backslash prod\_p\; p^,$ where $p$ runs over all prime numbers, and each $n\_p$ is zero, a natural number or infinity. Sometimes $v\_p(\backslash omega)$ is used instead of $n\_p$. If no $n\_p\; =\; \backslash infty$ and there are only a finite number of non-zero $n\_p$ then we recover the positive integers. Slightly less intuitively, if all $n\_p$ are $\backslash infty$, we get zero. Supernatural numbers extend beyond natural numbers by allowing the possibility of infinitely many prime factors, and by allowing any given prime to divide $\backslash omega$ "infinitely often," by taking that prime's corresponding exponent to be the symbol $\backslash infty$. There is no natural way to add supernatural numbers, but they can be multiplied, with $\backslash prod\_p\; p^\backslash cdot\backslash prod\_p\; p^=\backslash prod\_p\; p^$. Similarly, the notion of divisibility extends to the supernaturals with $\backslash omega\_1\backslash mid\backslash omega\_2$ if $v\_p(\backslash omega\_1)\backslash leq\; v\_p(\backslash omega\_2)$ for all $p$. The notion of the least common multiple and greatest common divisor can also be generalized for supernatural numbers, by defining : $\backslash displaystyle\; \backslash operatorname(\backslash )\; \backslash displaystyle\; =\backslash prod\_p\; p^$ : $\backslash displaystyle\; \backslash operatorname(\backslash )\; \backslash displaystyle\; =\backslash prod\_p\; p^$ With these definitions, the gcd or lcm of infinitely many natural numbers (or supernatural numbers) is a supernatural number. We can also extend the usual $p$-adic order functions to supernatural numbers by defining $v\_p(\backslash omega)=n\_p$ for each $p$ Supernatural numbers are used to define orders and indices of profinite groups and subgroups, in which case many of the theorems from finite group theory carry over exactly. They are used to encode the algebraic extensions of a finite field.〔Brawley & Schnibben (1989) pp.25-26〕 They are also used implicitly in many number-theoretical proofs, such as the density of the square-free integers and bounds for odd perfect numbers. ==See also== *profinite integer 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「Supernatural number」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.