Valuation theory
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In
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
(in particular in algebraic geometry or algebraic number theory), a valuation is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
on a field that provides a measure of size or multiplicity of elements of the field. It generalizes to
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
the notion of size inherent in consideration of the degree of a
pole Pole may refer to: Astronomy *Celestial pole, the projection of the planet Earth's axis of rotation onto the celestial sphere; also applies to the axis of rotation of other planets *Pole star, a visible star that is approximately aligned with the ...
or multiplicity of a
zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usual ...
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 mathematics, and in particular differential geometry and complex geometry, a complex analytic variety Complex analytic variety (or just variety) is sometimes required to be irreducible and (or) reduced or complex analytic space is a generali ...
in algebraic geometry. A field with a valuation on it is called a valued field.


Definition

One starts with the following objects: *a field and its
multiplicative group In mathematics and group theory, the term multiplicative group refers to one of the following concepts: *the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...
''K''×, *an abelian totally ordered group . The ordering and group law on are extended to the set by the rules * for all ∈ , * for all ∈ . Then a valuation of is any
map A map is a symbolic depiction emphasizing relationships between elements of some space, such as objects, regions, or themes. Many maps are static, fixed to paper or some other durable medium, while others are dynamic or interactive. Although ...
: which satisfies the following properties for all ''a'', ''b'' in ''K'': * if and only if , *, *, with equality if ''v''(''a'') ≠ ''v''(''b''). A valuation ''v'' is trivial if ''v''(''a'') = 0 for all ''a'' in ''K''×, otherwise it is non-trivial. The second property asserts that any valuation is a
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) w ...
. The third property is a version of the
triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but ...
on
metric spaces In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
adapted to an arbitrary Γ (see ''Multiplicative notation'' below). For valuations used in geometric applications, the first property implies that any non-empty
germ Germ or germs may refer to: Science * Germ (microorganism), an informal word for a pathogen * Germ cell, cell that gives rise to the gametes of an organism that reproduces sexually * Germ layer, a primary layer of cells that forms during embryo ...
of an analytic variety near a point contains that point. The valuation can be interpreted as the order of the
leading-order term The leading-order terms (or corrections) within a mathematical equation, expression or model are the terms with the largest order of magnitude.J.K.Hunter, ''Asymptotic Analysis and Singular Perturbation Theory'', 2004. http://www.math.ucdavis.edu ...
. The third property then corresponds to the order of a sum being the order of the larger term, unless the two terms have the same order, in which case they may cancel, in which case the sum may have larger order. For many applications, is an additive subgroup of the real numbers \R in which case ∞ can be interpreted as +∞ in the
extended real numbers In mathematics, the affinely extended real number system is obtained from the real number system \R by adding two infinity elements: +\infty and -\infty, where the infinities are treated as actual numbers. It is useful in describing the algebra on ...
; note that \min(a, +\infty) = \min(+\infty, a) = a for any real number ''a'', and thus +∞ is the unit under the binary operation of minimum. The real numbers (extended by +∞) with the operations of minimum and addition form a
semiring In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. The term rig is also used occasionally—this originated as a joke, suggesting that rigs ar ...
, called the min
tropical semiring In idempotent analysis, the tropical semiring is a semiring of extended real numbers with the operations of minimum (or maximum) and addition replacing the usual ("classical") operations of addition and multiplication, respectively. The tropical ...
, and a valuation ''v'' is almost a semiring homomorphism from ''K'' to the tropical semiring, except that the homomorphism property can fail when two elements with the same valuation are added together.


Multiplicative notation and absolute values

The concept was developed by Emil Artin in his book ''Geometric Algebra'' writing the group in multiplicative notation as : Emil Artinbr>''Geometric Algebra''
pages 47 to 49, via
Internet Archive The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music, ...
Instead of ∞, we adjoin a formal symbol ''O'' to Γ, with the ordering and group law extended by the rules * for all ∈ , * for all ∈ . Then a ''valuation'' of is any map : satisfying the following properties for all ''a'', ''b'' ∈ ''K'': * if and only if , *, *, with equality if . (Note that the directions of the inequalities are reversed from those in the additive notation.) If is a subgroup of the positive real numbers under multiplication, the last condition is the
ultrametric In mathematics, an ultrametric space is a metric space in which the triangle inequality is strengthened to d(x,z)\leq\max\left\. Sometimes the associated metric is also called a non-Archimedean metric or super-metric. Although some of the theorems ...
inequality, a stronger form of the
triangle inequality In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side. This statement permits the inclusion of degenerate triangles, but ...
, and is an absolute value. In this case, we may pass to the additive notation with value group \Gamma_+ \sub (\R, +) by taking . Each valuation on defines a corresponding linear preorder: . Conversely, given a "" satisfying the required properties, we can define valuation , with multiplication and ordering based on and .


Terminology

In this article, we use the terms defined above, in the additive notation. However, some authors use alternative terms: * our "valuation" (satisfying the ultrametric inequality) is called an "exponential valuation" or "non-Archimedean absolute value" or "ultrametric absolute value"; * our "absolute value" (satisfying the triangle inequality) is called a "valuation" or an "Archimedean absolute value".


Associated objects

There are several objects defined from a given valuation ; *the value group or valuation group = ''v''(''K''×), a subgroup of (though ''v'' is usually surjective so that = ); *the
valuation ring In abstract algebra, a valuation ring is an integral domain ''D'' such that for every element ''x'' of its field of fractions ''F'', at least one of ''x'' or ''x''−1 belongs to ''D''. Given a field ''F'', if ''D'' is a subring of ''F'' suc ...
''Rv'' is the set of ''a'' ∈ with ''v''(''a'') â‰¥ 0, *the prime ideal ''mv'' is the set of ''a'' ∈ ''K'' with ''v''(''a'') > 0 (it is in fact a
maximal ideal In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals c ...
of ''Rv''), *the residue field ''kv'' = ''Rv''/''mv'', *the place of associated to ''v'', the class of ''v'' under the equivalence defined below.


Basic properties


Equivalence of valuations

Two valuations ''v''1 and ''v''2 of with valuation group Γ1 and Γ2, respectively, are said to be equivalent if there is an order-preserving group isomorphism such that ''v''2(''a'') = Ï†(''v''1(''a'')) for all ''a'' in ''K''×. This is an equivalence relation. Two valuations of ''K'' are equivalent if and only if they have the same valuation ring. An equivalence class of valuations of a field is called a place. ''
Ostrowski's theorem In number theory, Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers \Q is equivalent to either the usual real absolute value or a -adic absolute value. Definitions Raisi ...
'' gives a complete classification of places of the field of
rational numbers In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
\Q: these are precisely the equivalence classes of valuations for the ''p''-adic completions of \Q.


Extension of valuations

Let ''v'' be a valuation of and let ''L'' be a field extension of . An extension of ''v'' (to ''L'') is a valuation ''w'' of ''L'' such that the
restriction Restriction, restrict or restrictor may refer to: Science and technology * restrict, a keyword in the C programming language used in pointer declarations * Restriction enzyme, a type of enzyme that cleaves genetic material Mathematics and logi ...
of ''w'' to is ''v''. The set of all such extensions is studied in the ramification theory of valuations. Let ''L''/''K'' be a
finite extension In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. The concept plays an important role in many parts of mathematics, including algebra and number theory &mdash ...
and let ''w'' be an extension of ''v'' to ''L''. The index of Γ''v'' in Γ''w'', e(''w''/''v'') =  ''w'' : Î“''v'' is called the reduced ramification index of ''w'' over ''v''. It satisfies e(''w''/''v'') â‰¤  'L'' : ''K''(the degree of the extension ''L''/''K''). The relative degree of ''w'' over ''v'' is defined to be ''f''(''w''/''v'') =  'Rw''/''mw'' : ''Rv''/''mv''(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, the ramification index of ''w'' over ''v'' is defined to be e(''w''/''v'')''pi'', where ''pi'' is the inseparable degree of the extension ''Rw''/''mw'' over ''Rv''/''mv''.


Complete valued fields

When the ordered abelian group is the additive group of the
integers An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
, the associated valuation is equivalent to an absolute value, and hence induces a metric on the field . If is complete with respect to this metric, then it is called a complete valued field. If ''K'' is not complete, one can use the valuation to construct its completion, as in the examples below, and different valuations can define different completion fields. In general, a valuation induces a
uniform structure In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifor ...
on , and is called a complete valued field if it is complete as a uniform space. There is a related property known as spherical completeness: it is equivalent to completeness if \Gamma = \Z, but stronger in general.


Examples


p-adic valuation

The most basic example is the -adic valuation ν''p'' associated to a prime integer ''p'', on the rational numbers K=\Q, with valuation ring R=\Z_, where \Z_ is the localization of \Z at the prime ideal (p) . The valuation group is the additive integers \Gamma = \Z. For an integer a \in R= \Z, the valuation ν''p''(''a'') measures the divisibility of ''a'' by powers of ''p'': : \nu_p(a) = \max\; and for a fraction, ν''p''(''a''/''b'') = ν''p''(''a'') − ν''p''(''b''). Writing this multiplicatively yields the -adic absolute value, which conventionally has as base 1/p = p^, so , a, _p := p^. The completion of \Q with respect to ν''p'' is the field \Q_p of p-adic numbers.


Order of vanishing

Let K = F(x), the rational functions on the affine line X = F1, and take a point ''a'' ∈ X. For a polynomial f(x) = a_k (xa)^k + a_(xa)^+\cdots+ a_n(xa)^n with a_k\neq 0, define ''v''''a''(''f'') = k, the order of vanishing at ''x'' = ''a''; and ''v''''a''(''f'' /''g'') = ''v''''a''(''f'') − ''v''''a''(''g''). Then the valuation ring ''R'' consists of rational functions with no pole at ''x'' = ''a'', and the completion is the formal Laurent series ring F((''x''−''a'')). This can be generalized to the field of
Puiseux series In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate. For example, the series : \begin x^ &+ 2x^ + x^ + 2x^ + x^ + x^5 + \cdots\\ &=x^+ 2x^ + x^ + 2x^ + x^ + ...
''K'' (fractional powers), the Levi-Civita field (its Cauchy completion), and the field of
Hahn series In mathematics, Hahn series (sometimes also known as Hahn–Mal'cev–Neumann series) are a type of formal infinite series. They are a generalization of Puiseux series (themselves a generalization of formal power series) and were first introduced ...
, with valuation in all cases returning the smallest exponent of ''t'' appearing in the series.


-adic valuation

Generalizing the previous examples, let be a principal ideal domain, be its
field of fractions In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded. The construction of the field of fractions is modeled on the relationship between the integral domain of integers and the field ...
, and be an
irreducible element In algebra, an irreducible element of a domain is a non-zero element that is not invertible (that is, is not a unit), and is not the product of two non-invertible elements. Relationship with prime elements Irreducible elements should not be confus ...
of . Since every principal ideal domain is a
unique factorization domain In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is ...
, every non-zero element ''a'' of can be written (essentially) uniquely as :a=\pi^p_1^p_2^\cdots p_n^ where the ''es are non-negative integers and the ''pi'' are irreducible elements of that are not associates of . In particular, the integer ''ea'' is uniquely determined by ''a''. The Ï€-adic valuation of ''K'' is then given by *v_\pi(0)=\infty *v_\pi(a/b)=e_a-e_b,\texta,b\in R, a, b\neq0. If Ï€' is another irreducible element of such that (Ï€') = (Ï€) (that is, they generate the same ideal in ''R''), then the Ï€-adic valuation and the Ï€'-adic valuation are equal. Thus, the Ï€-adic valuation can be called the ''P''-adic valuation, where ''P'' = (Ï€).


''P''-adic valuation on a Dedekind domain

The previous example can be generalized to Dedekind domains. Let be a Dedekind domain, its field of fractions, and let ''P'' be a non-zero prime ideal of . Then, the localization of at ''P'', denoted ''RP'', is a principal ideal domain whose field of fractions is . The construction of the previous section applied to the prime ideal ''PRP'' of ''RP'' yields the -adic valuation of .


Vector spaces over valuation fields

Suppose that ∪ is the set of non-negative real numbers under multiplication. Then we say that the valuation is non-discrete if its range (the valuation group) is infinite (and hence has an accumulation point at 0). 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'' ∈ ''K'' such that ''λ'' ∈ ''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. Also, ''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''. 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 f^(B). 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.


See also

* Discrete valuation * Euclidean valuation *
Field norm In mathematics, the (field) norm is a particular mapping defined in field theory, which maps elements of a larger field into a subfield. Formal definition Let ''K'' be a field and ''L'' a finite extension (and hence an algebraic extension) of ''K ...
*
Absolute value (algebra) In algebra, an absolute value (also called a valuation, magnitude, or norm, although " norm" usually refers to a specific kind of absolute value on a field) is a function which measures the "size" of elements in a field or integral domain. More ...


Notes


References

* *. A masterpiece on
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
written by one of the leading contributors. *Chapter VI of *


External links

* * * *{{MathWorld , title=Valuation , urlname=Valuation Algebraic geometry Field (mathematics)