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In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of size or multiplicity of elements of the field. They generalize to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in complex analysis, the degree of divisibility of a number by a prime number in number theory, and the geometrical concept of contact between two algebraic or analytic varieties in algebraic geometry. A field with a valuation on it is called a valued field. == Definition == To define the algebraic concept of valuation, the following objects are needed: *a field and its multiplicative subgroup ''K''×, *an abelian totally ordered group (which could also be given in multiplicative notation as ). The ordering and group law on are extended to the set 〔The symbol ∞ denotes an element not in , and has not any other meaning. Its properties are simply defined by axioms, as in every formal presentation of a mathematical theory.〕 by the rules * for all in , * for all α in . Then a valuation of is any map : which satisfies the following properties for all ''a'', ''b'' in ''K'': * if, and only if, , *, *, with equality if ''v''(''a'')≠''v''(''b''). Some authors use the term exponential valuation rather than "valuation". In this case the term "valuation" means "absolute value". A valuation ''v'' is called trivial (or the trivial valuation of ) if ''v''(''a'') = 0 for all ''a'' in ''K''×, otherwise it is called non-trivial. For valuations used in geometric applications, the first property implies that any non-empty germ of an analytic variety near a point contains that point. The second property asserts that ''any valuation is a group homomorphism'', while the third property is a translation of the triangle inequality from metric spaces to ordered groups. It is possible to give a dual definition of the same concept using the multiplicative notation for Γ: if, instead of ∞, an element ''O''〔As for the symbol ∞, ''O'' denotes an element not in Γ and has not any other meaning, its properties being again defined by axioms.〕 is given and the ordering and group law on Γ are extended by the rules * for all in , * for all α in , then a valuation of ''K'' is any map : satisfying the following properties for all ''a'', ''b'' in ''K'': * if, and only if, , *, *, with equality if ''v''(''a'')≠''v''(''b''). (Note that in this definition, the directions of the inequalities are reversed.) A valuation is commonly assumed to be surjective, since many arguments used in ordinary mathematical research involving those objects use preimages of unspecified elements of the ordered group contained in its codomain. Also, ''the first definition of valuation given is more frequently encountered in ordinary mathematical research'', thus it is the only one used in the following considerations and examples. 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「Valuation (algebra)」の詳細全文を読む スポンサード リンク
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