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:''For other uses, see Trace'' In mathematics, the field trace is a particular function defined with respect to a finite field extension ''L''/''K'', which is a ''K''-linear map from ''L'' to ''K''. ==Definition== Let ''K'' be a field and ''L'' a finite extension (and hence an algebraic extension) of ''K''. ''L'' can be viewed as a vector space over ''K''. Multiplication by α, an element of ''L'', :, is a ''K''-linear transformation of this vector space into itself. The ''trace'', TrL/K(α), is defined as the (linear algebra) trace of this linear transformation. For 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 and the coefficient above is one. More particularly, if ''L''/''K'' is a Galois extension and α is in ''L'', then the trace of α is the sum 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 trace」の詳細全文を読む スポンサード リンク
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