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field trace : ウィキペディア英語版
field trace
:''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'',
:m_\alpha:L\to L \text m_\alpha (x) = \alpha x,
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
:\operatorname_(\alpha)=()\sum_^n\sigma_j(\alpha).
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.
:\operatorname_(\alpha)=\sum_g(\alpha),
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)
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