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In mathematics, the (field) norm is a particular mapping defined in field theory, which maps elements of a larger field into a subfield. ==Formal definition== Let ''K'' be a field and ''L'' a finite extension (and hence an algebraic extension) of ''K''. The field ''L'' is then a finite dimensional vector space over ''K''. Multiplication by α, an element of ''L'', :, is a ''K''-linear transformation of this vector space into itself. The ''norm'', NL/K(α), is defined as the determinant of this linear transformation. For nonzero α in ''L'', let σ(α), ..., σ(α) be the roots (counted with multiplicity) of the minimal polynomial of α over ''K'' (in some extension field of ''L''), then :. If ''L''/''K'' is separable then each root appears only once in the product (the exponent () may still be greater than 1). More particularly, if ''L''/''K'' is a Galois extension and α is in ''L'', then the norm of α is the product of all the Galois conjugates of α, i.e. :, where Gal(''L''/''K'') denotes the Galois group of ''L''/''K''. 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「field norm」の詳細全文を読む スポンサード リンク
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