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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 cyclotomic polynomial ： ウィキペディア英語版 cyclotomic polynomial In mathematics, more specifically in algebra, the ''n''th cyclotomic polynomial, for any positive integer ''n'', is the unique irreducible polynomial with integer coefficients, which is a divisor of $x^n-1$ and is not a divisor of $x^k-1$ for any Its roots are the ''n''th primitive roots of unity $$ e^} , where ''k'' runs over the integers lower than ''n'' and coprime to ''n''. In other words, the ''n''th cyclotomic polynomial is equal to :$$ \Phi_n(x) = \prod_\stackrel \left(x-e^}\right) It may also be defined as the monic polynomial with integer coefficients, which is the minimal polynomial over the field of the rational numbers of any primitive ''n''th-root of unity ($e^$ is an example of such a root). Another important equation, linking the cyclotomic polynomials and primitive roots of unity, is the following one. :$x^n\; -\; 1\; =\backslash prod\_\; \backslash left(x-\; e^\; \backslash right)=\; \backslash prod\_\; \backslash prod\_\; \backslash left(x-\; e^\; \backslash right)\; =\backslash prod\_\; \backslash Phi\_(x)\; =\; \backslash prod\_\; \backslash Phi\_d(x)$ ==Examples== If ''n'' is a prime number then :$~\backslash Phi\_n(x)\; =\; 1+x+x^2+\backslash cdots+x^=\backslash sum\_^\; x^i.$ If ''n''=2''p'' where ''p'' is an odd prime number then :$~\backslash Phi\_(x)\; =\; 1-x+x^2-\backslash cdots+x^=\backslash sum\_^\; (-x)^i.$ For ''n'' up to 30, the cyclotomic polynomials are: :$~\backslash Phi\_1(x)\; =\; x\; -\; 1$ :$~\backslash Phi\_2(x)\; =\; x\; +\; 1$ :$~\backslash Phi\_3(x)\; =\; x^2\; +\; x\; +\; 1$ :$~\backslash Phi\_4(x)\; =\; x^2\; +\; 1$ :$~\backslash Phi\_5(x)\; =\; x^4\; +\; x^3\; +\; x^2\; +\; x\; +1$ :$~\backslash Phi\_6(x)\; =\; x^2\; -\; x\; +\; 1$ :$~\backslash Phi\_7(x)\; =\; x^6\; +\; x^5\; +\; x^4\; +\; x^3\; +\; x^2\; +\; x\; +\; 1$ :$~\backslash Phi\_8(x)\; =\; x^4\; +\; 1$ :$~\backslash Phi\_9(x)\; =\; x^6\; +\; x^3\; +\; 1$ :$~\backslash Phi\_(x)\; =\; x^4\; -\; x^3\; +\; x^2\; -\; x\; +\; 1$ :$~\backslash Phi\_(x)\; =\; x^\; +\; x^9\; +\; x^8\; +\; x^7\; +\; x^6\; +\; x^5\; +\; x^4\; +\; x^3\; +\; x^2\; +\; x\; +\; 1$ :$~\backslash Phi\_(x)\; =\; x^4\; -\; x^2\; +\; 1$ :$~\backslash Phi\_(x)\; =\; x^\; +\; x^\; +\; x^\; +\; x^9\; +\; x^8\; +\; x^7\; +\; x^6\; +\; x^5\; +\; x^4\; +\; x^3\; +\; x^2\; +\; x\; +\; 1$ :$~\backslash Phi\_(x)\; =\; x^6\; -\; x^5\; +\; x^4\; -\; x^3\; +\; x^2\; -\; x\; +\; 1$ :$~\backslash Phi\_(x)\; =\; x^8\; -\; x^7\; +\; x^5\; -\; x^4\; +\; x^3\; -\; x\; +\; 1$ :$~\backslash Phi\_(x)\; =\; x^8\; +\; 1$ :$~\backslash Phi\_(x)\; =\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^9\; +\; x^8\; +\; x^7\; +\; x^6\; +\; x^5\; +\; x^4\; +\; x^3\; +\; x^2\; +\; x\; +\; 1$ :$~\backslash Phi\_(x)\; =\; x^6\; -\; x^3\; +\; 1$ :$~\backslash Phi\_(x)\; =\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^9\; +\; x^8\; +\; x^7\; +\; x^6\; +\; x^5\; +\; x^4\; +\; x^3\; +\; x^2\; +\; x\; +\; 1$ :$~\backslash Phi\_(x)\; =\; x^8\; -\; x^6\; +\; x^4\; -\; x^2\; +\; 1$ :$~\backslash Phi\_(x)\; =\; x^\; -\; x^\; +\; x^9\; -\; x^8\; +\; x^6\; -\; x^4\; +\; x^3\; -\; x\; +\; 1$ :$~\backslash Phi\_(x)\; =\; x^\; -\; x^9\; +\; x^8\; -\; x^7\; +\; x^6\; -\; x^5\; +\; x^4\; -\; x^3\; +\; x^2\; -\; x\; +\; 1$ :$~\backslash Phi\_(x)\; =\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^9\; +\; x^8\; +\; x^7\; +\; x^6\; +\; x^5\; +\; x^4\; +\; x^3\; +\; x^2\; +\; x\; +\; 1$ :$~\backslash Phi\_(x)\; =\; x^8\; -\; x^4\; +\; 1$ :$~\backslash Phi\_(x)\; =\; x^\; +\; x^\; +\; x^\; +\; x^5\; +\; 1$ :$~\backslash Phi\_(x)\; =\; x^\; -\; x^\; +\; x^\; -\; x^9\; +\; x^8\; -\; x^7\; +\; x^6\; -\; x^5\; +\; x^4\; -\; x^3\; +\; x^2\; -\; x\; +\; 1$ :$~\backslash Phi\_(x)\; =\; x^\; +\; x^9\; +\; 1$ :$~\backslash Phi\_(x)\; =\; x^\; -\; x^\; +\; x^8\; -\; x^6\; +\; x^4\; -\; x^2\; +\; 1$ :$~\backslash Phi\_(x)\; =\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^\; +\; x^9\; +\; x^8\; +\; x^7\; +\; x^6\; +\; x^5\; +\; x^4\; +\; x^3\; +\; x^2\; +\; x\; +\; 1$ :$~\backslash Phi\_(x)\; =\; x^8\; +\; x^7\; -\; x^5\; -\; x^4\; -\; x^3\; +\; x\; +\; 1$ The case of the 105th cyclotomic polynomial is interesting because 105 is the lowest integer that is the product of three distinct odd prime numbers and this polynomial is the first one that has a coefficient greater than 1: :$\backslash begin$ \Phi_(x) = & \; x^ + x^ + x^ - x^ - x^ - 2 x^ - x^ - x^ + x^ + x^ + x^ \\ & + x^ - x^ - x^ - x^ - x^ - x^ + x^ + x^ + x^ \\ & + x^ - x^9 - x^8 - 2 x^7 - x^6 - x^5 + x^2 + x + 1 \end 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「cyclotomic polynomial」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.