翻訳と辞書 Words near each other ・ Eisenmenger's syndrome ・ Eisenschmitt ・ Eisenson ・ Eisenstadt ・ Eisenstadt (disambiguation) ・ Eisenstadt (surname) ・ Eisenstadt Cathedral ・ Eisenstadt v. Baird ・ Eisenstadt-Umgebung District ・ Eisenstein ・ Eisenstein (film) ・ Eisenstein (Ore Mountains) ・ Eisenstein (surname) ・ Eisenstein ideal ・ Eisenstein in Guanajuato ・ Eisenstein integer ・ Eisenstein integral ・ Eisenstein on Disney ・ Eisenstein prime ・ Eisenstein reciprocity ・ Eisenstein series ・ Eisenstein sum ・ Eisenstein triple ・ Eisenstein's criterion ・ Eisenstein's theorem ・ Eisenstein, Wisconsin ・ Eisenstein–Kronecker number ・ Eisentraut's mouse shrew ・ Eisentraut's pipistrelle ・ Eisentraut's shrew Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Eisenstein integer ： ウィキペディア英語版 Eisenstein integer In mathematics, Eisenstein integers (named after Gotthold Eisenstein), also known〔 as Eulerian integers (after Leonhard Euler), are complex numbers of the form :$z\; =\; a\; +\; b\backslash omega\; ,$ where ''a'' and ''b'' are integers and :$\backslash omega\; =\; \backslash frac(-1\; +\; i\backslash sqrt\; 3)\; =\; e^$ is a primitive (non-real) cube root of unity. The Eisenstein integers form a triangular lattice in the complex plane, in contrast with the Gaussian integers, which form a square lattice in the complex plane. ==Properties== The Eisenstein integers form a commutative ring of algebraic integers in the algebraic number field Q(ω) — the third cyclotomic field. To see that the Eisenstein integers are algebraic integers note that each ''z'' = ''a'' + ''b''ω is a root of the monic polynomial :$z^2\; -\; (2a\; -\; b)z\; +\; (a^2\; -\; ab\; +\; b^2).\; \backslash ,\backslash !$ In particular, ω satisfies the equation :$\backslash omega^2\; +\; \backslash omega\; +\; 1\; =\; 0.\; \backslash ,\backslash !$ The product of two Eisenstein integers $a+b\backslash omega$ and $c+d\backslash omega$ is given explicitly by :$(a+b\backslash omega)\; \backslash cdot\; (c+d\backslash omega)=(ac-bd)+(bc+ad-bd)\backslash omega.\; \backslash ,\backslash !$ The norm of an Eisenstein integer is just the square of its modulus and is given by :$|a+b\backslash omega|^2\; =\; a^2\; -\; ab\; +\; b^2.\; \backslash ,\backslash !$ Thus the norm of an Eisenstein integer is always an ordinary (rational) integer. Since :$4a^2-4ab+4b^2=(2a-b)^2+3b^2,\; \backslash ,\backslash !$ the norm of a nonzero Eisenstein integer is positive. The group of units in the ring of Eisenstein integers is the cyclic group formed by the sixth roots of unity in the complex plane. Specifically, they are : These are just the Eisenstein integers of norm one. 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「Eisenstein integer」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.