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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 Monogenic field ： ウィキペディア英語版 Monogenic field In mathematics, a monogenic field is an algebraic number field ''K'' for which there exists an element ''a'' such that the ring of integers ''O''''K'' is the polynomial ring Z(). The powers of such an element ''a'' constitute a power integral basis. In a monogenic field ''K'', the field discriminant of ''K'' is equal to the discriminant of the minimal polynomial of α. ==Examples== Examples of monogenic fields include: * Quadratic fields: : if $K\; =\; \backslash mathbf(\backslash sqrt\; d)$ with $d$ a square-free integer, then $O\_K\; =\; \backslash mathbf()$ where $a\; =\; (1+\backslash sqrt\; d)/2$ if ''d''≡1 (mod 4) and $a\; =\; \backslash sqrt\; d$ if ''d'' ≡ 2 or 3 (mod 4). * Cyclotomic fields: : if $K\; =\; \backslash mathbf(\backslash zeta)$ with $\backslash zeta$ a root of unity, then $O\_K\; =\; \backslash mathbf().$ Also the maximal real subfield $\backslash mathbf(\backslash zeta)^\; =\; \backslash mathbf(\backslash zeta\; +\; \backslash zeta^)$ is monogenic, with ring of integers $\backslash mathbf().$ While all quadratic fields are monogenic, already among cubic fields there are many that are not monogenic. The first example of a non-monogenic number field that was found is the cubic field generated by a root of the polynomial $X^3\; -\; X^2\; -\; 2X\; -\; 8$, due to Richard Dedekind. 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「Monogenic field」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.