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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 and 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 : is the unique group homomorphism that satisfies : for all nonzero prime ideals of ''B'', where is the prime ideal of ''A'' lying below . Alternatively, for any one can equivalently define to be the fractional ideal of ''A'' generated by the set of field norms of elements of ''B''. For , one has , where . The ideal norm of a principal ideal is thus compatible with the field norm of an element: Let be a Galois extension of number fields with rings of integers . Then the preceding applies with , and for any we have : which is an element of . The notation is sometimes shortened to , an abuse of notation that is compatible with also writing for the field norm, as noted above. In the case , it is reasonable to use positive rational numbers as the range for since has trivial ideal class group and unit group , thus each nonzero fractional ideal of is generated by a uniquely determined positive rational number. Under this convention the relative norm from down to 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
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