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In mathematics, the residue field is a basic construction in commutative algebra. If ''R'' is a commutative ring and ''m'' is a maximal ideal, then the residue field is the quotient ring ''k'' = ''R''/''m'', which is a field. Frequently, ''R'' is a local ring and ''m'' is then its unique maximal ideal. This construction is applied in algebraic geometry, where to every point ''x'' of a scheme ''X'' one associates its residue field ''k''(''x''). One can say a little loosely that the residue field of a point of an abstract algebraic variety is the 'natural domain' for the coordinates of the point. ==Definition== Suppose that ''R'' is a commutative local ring, with the maximal ideal ''m''. Then the residue field is the quotient ring ''R''/''m''. Now suppose that ''X'' is a scheme and ''x'' is a point of ''X''. By the definition of scheme, we may find an affine neighbourhood ''U'' = Spec(''A''), with ''A'' some commutative ring. Considered in the neighbourhood ''U'', the point ''x'' corresponds to a prime ideal ''p'' ⊂ ''A'' (see Zariski topology). The ''local ring'' of ''X'' in ''x'' is by definition the localization ''R'' = ''Ap'', with the maximal ideal ''m'' = ''p·Ap''. Applying the construction above, we obtain the residue field of the point ''x'' : :''k''(''x'') := ''A''''p'' / ''p''·''A''''p''. One can prove that this definition does not depend on the choice of the affine neighbourhood ''U''.〔Intuitively, the residue field of a point is a local invariant. Axioms of schemes are set up in such a way as to assure the compatibility between various affine open neighborhoods of a point, which implies the statement.〕 A point is called ''K''-rational for a certain field ''K'', if ''k''(''x'') ⊂ ''K''.〔Görtz, Ulrich and Wedhorn, Torsten. ''Algebraic Geometry: Part 1: Schemes'' (2010) Vieweg+Teubner Verlag.〕 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「Residue field」の詳細全文を読む スポンサード リンク
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