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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 Ideal norm ： ウィキペディア英語版 Ideal norm In commutative algebra, the norm of an ideal is a generalization of a norm of an element in the field extension. It is particularly important in number theory since it measures the size of an ideal of a complicated number ring in terms of an ideal in a less complicated ring. When the less complicated number ring is taken to be the ring of integers, Z, then the norm of a nonzero ideal ''I'' of a number ring ''R'' is simply the size of the finite quotient ring ''R''/''I''. == Relative norm == Let ''A'' be a Dedekind domain with field of fractions ''K'' and integral closure of ''B'' in a finite separable extension ''L'' of ''K''. (this implies that ''B'' is also a Dedekind domain.) Let $\backslash mathcal\_A$ and $\backslash mathcal\_B$ be the ideal groups of ''A'' and ''B'', respectively (i.e., the sets of nonzero fractional ideals.) Following the technique developed by Jean-Pierre Serre, the norm map :$N\_\backslash colon\; \backslash mathcal\_B\; \backslash to\; \backslash mathcal\_A$ is the unique group homomorphism that satisfies :$N\_(\backslash mathfrak\; q)\; =\; \backslash mathfrak^$ for all nonzero prime ideals $\backslash mathfrak\; q$ of ''B'', where $\backslash mathfrak\; p\; =\; \backslash mathfrak\; q\backslash cap\; A$ is the prime ideal of ''A'' lying below $\backslash mathfrak\; q$. Alternatively, for any $\backslash mathfrak\; b\backslash in\backslash mathcal\_B$ one can equivalently define $N\_(\backslash mathfrak)$ to be the fractional ideal of ''A'' generated by the set $\backslash \; \backslash \}$ of field norms of elements of ''B''. For $\backslash mathfrak\; a\; \backslash in\; \backslash mathcal\_A$, one has $N\_(\backslash mathfrak\; a\; B)\; =\; \backslash mathfrak\; a^n$, where $n\; =\; (:\; K)$. The ideal norm of a principal ideal is thus compatible with the field norm of an element: $N\_(xB)\; =\; N\_(x)A.$ Let $L/K$ be a Galois extension of number fields with rings of integers $\backslash mathcal\_K\backslash subset\; \backslash mathcal\_L$. Then the preceding applies with $A\; =\; \backslash mathcal\_K,\; B\; =\; \backslash mathcal\_L$, and for any $\backslash mathfrak\; b\backslash in\backslash mathcal\_$ we have :$N\_\_K\}(\backslash mathfrak\; b)=\backslash mathcal\_K\; \backslash cap\backslash prod\_\; \backslash sigma\; (\backslash mathfrak\; b),$ which is an element of $\backslash mathcal\_$. The notation $N\_\_K\}$ is sometimes shortened to $N\_$, an abuse of notation that is compatible with also writing $N\_$ for the field norm, as noted above. In the case $K=\backslash mathbb$, it is reasonable to use positive rational numbers as the range for $N\_\}\backslash ,$ since $\backslash mathbb$ has trivial ideal class group and unit group $\backslash$, thus each nonzero fractional ideal of $\backslash mathbb$ is generated by a uniquely determined positive rational number. Under this convention the relative norm from $L$ down to $K=\backslash mathbb$ coincides with the absolute norm defined below. 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「Ideal norm」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.