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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 Fundamental unit (number theory) ： ウィキペディア英語版 Fundamental unit (number theory) In algebraic number theory, a fundamental unit is a generator (modulo the roots of unity) for the unit group of the ring of integers of a number field, when that group has rank 1 (i.e. when the unit group modulo its torsion subgroup is infinite cyclic). Dirichlet's unit theorem shows that the unit group has rank 1 exactly when the number field is a real quadratic field, a complex cubic field, or a totally imaginary quartic field. When the unit group has rank ≥ 1, a basis of it modulo its torsion is called a fundamental system of units. Some authors use the term fundamental unit to mean any element of a fundamental system of units, not restricting to the case of rank 1 (e.g. ). ==Real quadratic fields== For the real quadratic field $K=\backslash mathbf(\backslash sqrt)$ (with ''d'' square-free), the fundamental unit ε is commonly normalized so that (as a real number). Then it is uniquely characterized as the minimal unit among those that are greater than 1. If Δ denotes the discriminant of ''K'', then the fundamental unit is :$\backslash varepsilon=\backslash frac$ where (''a'', ''b'') is the smallest solution to :$x^2-\backslash Delta\; y^2=\backslash pm4$ in positive integers. This equation is basically Pell's equation or the negative Pell equation and its solutions can be obtained similarly using the continued fraction expansion of $\backslash sqrt$. Whether or not ''x''2 − Δ''y''2 = −4 has a solution determines whether or not the class group of ''K'' is the same as its narrow class group, or equivalently, whether or not there is a unit of norm −1 in ''K''. This equation is known to have a solution if, and only if, the period of the continued fraction expansion of $\backslash sqrt$ is odd. A simpler relation can be obtained using congruences: if Δ is divisible by a prime that is congruent to 3 modulo 4, then ''K'' does not have a unit of norm −1. However, the converse does not hold as shown by the example ''d'' = 34. In the early 1990s, Peter Stevenhagen proposed a probabilistic model that led him to a conjecture on how often the converse fails. Specifically, if ''D''(''X'') is the number of real quadratic fields whose discriminant Δ < ''X'' is not divisible by a prime congruent to 3 modulo 4 and ''D''−(''X'') is those who have a unit of norm −1, then :$\backslash lim\_\backslash frac=1-\backslash prod\_\backslash right).$ In other words, the converse fails about 42% of the time. As of March 2012, a recent result towards this conjecture was provided by Étienne Fouvry and Jürgen Klüners who show that the converse fails between 33% and 59% of the time. 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「Fundamental unit (number theory)」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.