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| In [[algebra]] (in particular in [[algebraic geometry]] or [[algebraic number field|algebraic number theory]]), a '''valuation''' is a [[Function (mathematics)|function]] on a [[Field (mathematics)|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 (complex analysis)|pole]] or [[Multiplicity (mathematics)|multiplicity]] of a [[zero (complex analysis)|zero]] in complex analysis, the degree of divisibility of a number by a prime number in number theory, and the geometrical concept of [[contact (geometry)|contact]] between two [[algebraic variety|algebraic]] or [[analytic variety|analytic varieties]] in algebraic geometry. A field with a valuation on it is called a '''valued field'''.
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| == Definition ==
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| To define the algebraic concept of valuation, the following objects are needed:
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| *a [[field (mathematics)|field]] ''K'' and its [[multiplicative subgroup]] ''K''<sup>×</sub>,
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| *an [[abelian group|abelian]] [[totally ordered group]] (Γ, +, ≥) (which could also be given in [[multiplicative notation]] as (Γ, ·, ≥)).
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| The ordering and [[group law]] on Γ are extended to the [[set (mathematics)|set]] Γ∪{∞}<ref>The symbol ∞ denotes an [[Element (mathematics)|element]] not in Γ, and has not any other meaning. Its properties are simply defined by [[axiom]]s, as in every [[Formalism (mathematics)|formal presentation]] of a [[mathematical theory]].</ref> by the rules
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| *∞ ≥ α for all α in Γ,
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| *{{nowrap|∞ + α {{=}} α + ∞ {{=}} ∞}} for all α in Γ.
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| Then a '''valuation of ''K''''' is any [[Map (mathematics)|map]]
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| :{{nowrap|''v'' : ''K'' → Γ∪{∞}}}
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| which satisfies the following properties for all ''a'', ''b'' in ''K'':
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| *{{nowrap|''v''(''a'') {{=}} ∞}} if, and only if, {{nowrap|''a'' {{=}} 0}},
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| *{{nowrap|''v''(''ab'') {{=}} ''v''(''a'') + ''v''(''b'')}},
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| *{{nowrap|''v''(''a'' + ''b'') ≥ min(''v''(''a''), ''v''(''b''))}}.
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| Some authors use the term '''exponential valuation''' rather than "valuation". In this case the term "valuation" means "[[absolute value (algebra)|absolute value]]".
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| A valuation ''v'' is called '''trivial''' (or the '''trivial valuation of ''K''''') if ''v''(''a'') = 0 for all ''a'' in ''K''<sup>×</sup>, otherwise it is called '''non-trivial'''.
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| For valuations used in [[analytic geometry|geometric]] applications, the first property implies that any non-empty [[germ (mathematics)|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.
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| It is possible to give a [[Duality (mathematics)|dual]] definition of the same concept using the multiplicative notation for Γ: if, instead of ∞, an element ''O''<ref>As for the symbol ∞, ''O'' denotes an [[Element (mathematics)|element]] not in Γ and has not any other meaning, its properties being again defined by [[axiom]]s.</ref> is given and the ordering and group law on Γ are extended by the rules
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| *{{nowrap|''O'' ≤ α}} for all α in Γ,
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| *{{nowrap|''O'' · α {{=}} α · ''O'' {{=}} ''O''}} for all α in Γ,
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| then a '''valuation of ''K''''' is any map
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| :{{nowrap|''v'' : ''K'' → Γ∪{''O''}}}
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| satisfying the following properties for all ''a'', ''b'' in ''K'':
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| *{{nowrap|''v''(''a'') {{=}} ''O''}} if, and only if, {{nowrap|''a'' {{=}} 0}},
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| *{{nowrap|''v''(''ab'') {{=}} ''v''(''a'') · ''v''(''b'')}},
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| *{{nowrap|''v''(''a'' + ''b'') ≤ max(''v''(''a''), ''v''(''b''))}}.
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| (Note that in this definition, the directions of the inequalities are reversed.)
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| A valuation is commonly assumed to be [[surjective]], since many arguments used in ordinary mathematical research involving those objects use [[preimage]]s 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.
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| ===Associated objects===
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| If {{nowrap|''v'' : ''K'' → Γ∪{∞}}} is a valuation, then there are several objects that can be defined from it:
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| *the '''value group of ''v''''' (or '''valuation group of ''v'''''), denoted Γ<sub>''v''</sub>, is ''v''(''K''<sup>×</sup>), it is a subgroup of Γ,
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| *the '''valuation ring of ''v''''', denoted ''R<sub>v</sub>'' is the set of elements ''a'' of ''K'' such that ''v''(''a'') ≥ 0, it is a [[valuation ring]],
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| *the '''prime ideal of ''v''''' (or the '''maximal ideal of ''v'''''), denoted ''m<sub>v</sub>'' is the set of elements ''a'' of ''K'' such that ''v''(''a'') > 0, it is a [[maximal ideal]] of ''R<sub>v</sub>'',
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| *the '''residue field of ''v''''', denoted ''k<sub>v</sub>'' is ''R<sub>v</sub>''/''m<sub>v</sub>'', it is a field.
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| ==Basic properties==
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| ==={{anchor|equivalence}} Equivalence of valuations ===
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| Two valuations ''v''<sub>1</sub> and ''v''<sub>2</sub> of ''K'' with valuation group Γ<sub>1</sub> and Γ<sub>2</sub>, respectively, are said to be '''equivalent''' if there is an order-preserving [[group isomorphism]] {{nowrap|φ : Γ<sub>1</sub> → Γ<sub>2</sub>}} such that ''v''<sub>2</sub>(''a'') = φ(''v''<sub>1</sub>(''a'')) for all ''a'' in ''K''<sup>×</sup>. This is an [[equivalence relation]].
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| Two valuations of ''K'' are equivalent if, and only if, they have the same valuation ring.
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| An [[equivalence class]] of valuations of a field is called a '''place'''. ''[[Ostrowski's theorem]]'' gives a complete classification of places of the field of [[rational numbers]] '''Q''': these are precisely the equivalence classes of valuations for the [[p-adic number|''p''-adic]] [[complete space|completions]] of '''Q'''.
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| ==={{anchor|extensions}} Extension of valuations===
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| Let ''v'' be a valuation of ''K'' and let ''L'' be a [[field extension]] of ''K''. An '''extension of ''v''''' (to ''L'') is a valuation ''w'' of ''L'' such that the [[function restriction|restriction]] of ''w'' to ''K'' is ''v''. The set of all such extensions is studied in the [[ramification theory of valuations]].
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| Let ''L''/''K'' be a [[finite extension]] and let ''w'' be an extension of ''v'' to ''L''. The [[index of a subgroup|index]] of Γ<sub>''v''</sub> in Γ<sub>''w''</sub>, e(''w''/''v'') = [Γ<sub>''w''</sub> : Γ<sub>''v''</sub>], is called the '''reduced ramification index''' of ''w'' over ''v''. It satisfies e(''w''/''v'') ≤ [''L'' : ''K''] (the [[degree of a field extension|degree]] of the extension ''L''/''K''). The '''relative degree''' of ''w'' over ''v'' is defined to be ''f''(''w''/''v'') = [''R<sub>w</sub>''/''m<sub>w</sub>'' : ''R<sub>v</sub>''/''m<sub>v</sub>''] (the degree of the extension of residue fields). It is also less than or equal to the degree of ''L''/''K''. When ''L''/''K'' is [[separable extension|separable]], the '''ramification index''' of ''w'' over ''v'' is defined to be e(''w''/''v'')''p<sup>i</sup>'', where ''p<sup>i</sup>'' is the [[inseparable degree]] of the extension ''R<sub>w</sub>''/''m<sub>w</sub>'' over ''R<sub>v</sub>''/''m<sub>v</sub>''.
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| === Complete valued fields ===<!-- Complete valued field redirects here-->
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| When the ordered abelian group Γ is the additive group of the [[integers]], the associated absolute value induces a [[Metric (mathematics)|metric]] on the field ''K''. If ''K'' is [[Complete metric space|complete]] with respect to this metric, then it is called a '''complete valued field'''. There is a related property known as [[Spherically complete field|spherical completeness]]: it is equivalent to completeness if Γ is '''Z''', but stronger in general.
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| == Examples ==
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| === π-adic valuation ===
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| Let ''R'' be a [[principal ideal domain]], ''K'' be its [[field of fractions]], and π be an [[irreducible element]] of ''R''. Since every principal ideal domain is a [[unique factorization domain]], every non-zero element ''a'' of ''R'' can be written (essentially) uniquely as
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| :<math>a=\pi^{e_a}p_1^{e_1}p_2^{e_2}\cdots p_n^{e_n}</math>
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| where the ''e'''s are non-negative integers and the ''p<sub>i</sub>'' are irreducible elements of ''R'' that are not [[Associate (ring theory)|associates]] of π. In particular, the integer ''e<sub>a</sub>'' is uniquely determined by ''a''.
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| The '''π-adic valuation of ''K''''' is then given by | |
| *<math>v_\pi(0)=\infty</math>
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| *<math>v_\pi(a/b)=e_a-e_b,\text{ for }a,b\in R, a, b\neq0.</math>
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| If π' is another irreducible element of ''R'' such that (π') = (π) (that is, they generate the same ideal in ''R''), then the π-adic valuation and the π'-adic valuation are equal. Thus, the π-adic valuation can be called the ''P''-adic valuation, where ''P'' = (π).
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| When ''R'' is the [[ring of integers]] '''Z''', then ''K'' is the [[rational number]]s '''Q''', and π is some [[prime number]] ''p'' (or its negative). The π-adic valuation obtained is the [[additive p-adic valuation|''p''-adic valuation]] on '''Q'''.
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| ==={{anchor|DedekindDomain}} ''P''-adic valuation on a Dedekind domain===
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| The previous example can be generalized to [[Dedekind domains]]. Let ''R'' be a Dedekind domain, ''K'' its field of fractions, and let ''P'' be a non-zero prime ideal of ''R''. Then, the [[localization of a ring|localization]] of ''R'' at ''P'', denoted ''R<sub>P</sub>'', is a principal ideal domain whose field of fractions is ''K''. The construction of the previous section applied to the prime ideal ''PR<sub>P</sub>'' of ''R<sub>P</sub>'' yields the '''''P''-adic valuation of ''K'''''.
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| === Geometric notion of contact ===
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| Let <math> \mathfrak{R} = \mathbb{C}[x,y] </math> be the ring of polynomials of two variables over the complex field, <math> \mathbb{K} = \mathbb{C}(x,y) </math> be the field of rational functions over the same field, and consider the (convergent) [[power series]]
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| :<math> f(x,y) = y - \sum_{n=3}^{\infty} x^n \in \mathbb{C}\{x,y\}</math>
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| whose [[zero set]], the analytic variety ''<math> V_f</math>'', can be parametrized by one [[coordinate system|coordinate]] ''<math>t</math>'' as follows
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| :<math> V_f = \{(x,y)\in\mathbb{C}^2\mid f(x,y) = 0\} = \Bigl\{ (x,y)\in\mathbb{C}^2\mid(x,y) = \Bigl(t,\sum_{n=3}^{\infty}t^n\Bigr)\Bigr\}</math>
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| It is possible to define a map <math> v\colon \mathbb{C}[x,y] \rightarrow \mathbb{Z} </math> as ''the value of the [[Power series#Order of a power series|order]] of the formal power series in the variable ''<math>t</math>'' obtained by [[Function (mathematics)#Restrictions and extensions|restriction]] of any polynomial ''<math>P</math>'' in <math> \mathbb{C}[x,y] </math> to the points of the set ''<math> V_f</math>''
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| :<math>
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| v(P) = \mathrm{ord}_t\left(P|_{V_f}\right) = \mathrm{ord}_t \Bigl(P\Bigl(t,\sum_{n=3}^{+\infty}t^n\Bigr)\Bigr) \quad \forall P\in \mathbb{C}[x,y]
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| </math>
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| It is also possible to extend the map <math>v</math> from its original ring of definition to the whole field <math> \mathbb{C}(x,y) </math> as follows
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| :<math>
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| v(P/Q) =
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| \begin{cases}
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| v(P) - v(Q) & P/Q \in {\mathbb{C}(x,y)}^* \\
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| \infty & P \equiv 0 \in \mathbb{C}(x,y)
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| \end{cases}
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| </math>
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| As the power series ''<math>f</math>'' is not a [[polynomial]], it is easy to prove that the extended map ''<math>v</math>'' is a valuation: the value ''<math>v</math>''<math>(P)</math>'' is called ''[[intersection number]] between the [[curve]]s (1-[[dimension]]al [[analytic variety|analytic varieties]]) ''<math> V_P</math>'' and ''<math>V_f</math>''. As an example, the computation of some intersection numbers follows:
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| :<math>
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| \begin{array}{l}
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| v(x) = \mathrm{ord}_t(t) = 1 \\
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| v(x^6-y^2)=\mathrm{ord}_t(t^6-t^6-2t^7-3t^8-\cdots)=\mathrm{ord}_t (-2t^7-3t^8-\cdots)=7 \\
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| v\left(\frac{x^6 - y^2}{x}\right)= \mathrm{ord}_t (-2t^7-3t^8-\cdots) - \mathrm{ord}_t(t) = 7 - 1 = 6
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| \end{array}
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| </math>
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| ==Vector spaces over valuation fields==
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| Suppose that Γ is the set of non-negative real numbers. Then we say that the valuation is '''non-discrete''' if its range is not finite.
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| Suppose that ''X'' is a vector space over ''K'' and that ''A'' and ''B'' are subsets of ''X''. Then we say that '''''A'' absorbs ''B''''' if there exists a ''α'' in ''K'' such that ''λ'' in ''K'' and ''|λ| ≥ |α|'' implies that ''B ⊆ λ A''. ''A'' is called '''radial''' or '''absorbing''' if ''A'' absorbs every finite subset of ''X''. Radial subsets of ''X'' are invariant under finite intersection. And ''A'' is called '''circled''' if ''λ'' in ''K'' and ''|λ| ≥ |α|'' implies ''λ A ⊆ A''. The set of circled subsets of ''L'' is invariant under arbitrary intersections. The '''circled hull''' of ''A'' is the intersection of all circled subsets of ''X'' containing ''A''.
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| Suppose that ''X'' and ''Y'' are vector spaces over a non-discrete valuation field ''K'', let ''A ⊆ X'', ''B ⊆ Y'', and let ''f : X → Y'' be a linear map. If ''B'' is circled or radial then so is <math>f^{-1}(B)</math>. If ''A'' is circled then so is ''f(A)'' but if ''A'' is radial then ''f(A)'' will be radial under the additional condition that ''f'' is surjective.
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| == See also ==
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| *[[Discrete valuation]]
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| *[[Euclidean valuation]]
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| *[[Valuation (measure theory)]]
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| *[[Valuation ring]]
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| ==Notes==
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| {{reflist|30em}}
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| == References ==
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| * {{citation | last=Efrat | first=Ido | title=Valuations, orderings, and Milnor ''K''-theory | series=Mathematical Surveys and Monographs | volume=124 | location=Providence, RI | publisher=[[American Mathematical Society]] | year=2006 | isbn=0-8218-4041-X | zbl=1103.12002 }}
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| *{{Citation
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| | last = Jacobson
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| | first = Nathan
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| | author-link = Nathan Jacobson
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| | title = Basic algebra II
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| | place = New York
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| | publisher = [[W. H. Freeman|W. H. Freeman and Company]]
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| | origyear = 1980
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| | year = 1989
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| | edition = 2nd
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| | chapter = Valuations: paragraph 6 of chapter 9
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| | zbl = 0694.16001
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| | isbn = 0-7167-1933-9}}. A masterpiece on [[algebra]] written by one of the leading contributors.
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| *Chapter VI of {{Citation
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| | last=Zariski
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| | first=Oscar
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| | author-link=Oscar Zariski
| |
| | last2=Samuel
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| | first2=Pierre
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| | author2-link=Pierre Samuel
| |
| | title=Commutative algebra, Volume II
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| | publisher=Springer-Verlag
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| | location=New York, Heidelberg
| |
| | series=[[Graduate Texts in Mathematics]]
| |
| | volume=29
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| | year=1976
| |
| | origyear=1960
| |
| | isbn=978-0-387-90171-8 | zbl=0322.13001
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| }}
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| * {{Cite isbn|0387987266|pages = 10 - 11}}
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| == External links ==
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| *{{springer
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| | title= Valuation
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| | id= V/v096010
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| | last= Danilov
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| | first= V.I.
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| }}
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| *{{PlanetMath|urlname=DiscreteValuation|title=Discrete valuation}}
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| *{{PlanetMath|urlname=Valuation|title=Valuation}}
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| *{{MathWorld |title=Valuation |urlname=Valuation}}
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| [[Category:Algebraic geometry]]
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| [[Category:Field theory]]
| |
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