Valuation (algebra)
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algebra Algebra () is one of the areas of mathematics, 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 mathem ...
(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. 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. Promi ...
the notion of size inherent in consideration of the degree of a pole 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, usu ...
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 ''K''×, *an
abelian Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a grou ...
totally ordered group . The ordering and
group law In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. The ...
on are extended to the set by the rules * for all ∈ , * for all ∈ . 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''). 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. 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, bu ...
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 setti ...
adapted to an arbitrary Γ (see ''Multiplicative notation'' below). 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 valuation can be interpreted as the order of the leading-order term. 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; 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 a ...
, 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 tropic ...
, 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 Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing ...
in his book ''Geometric Algebra'' writing the group in multiplicative notation as :
Emil Artin Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent. Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing ...
br>''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 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, bu ...
, 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 ''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 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 In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two g ...
such that ''v''2(''a'') = Ï†(''v''1(''a'')) for all ''a'' in ''K''×. This is an
equivalence relation In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relatio ...
. Two valuations of ''K'' are equivalent if and only if they have the same valuation ring. An
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements ...
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 completions of \Q.


Extension of valuations

Let ''v'' be a valuation of and let ''L'' be a
field extension In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
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 log ...
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 and let ''w'' be an extension of ''v'' to ''L''. The
index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ...
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 Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathemati ...
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 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 ''K'' (fractional powers), the Levi-Civita field (its Cauchy completion), and the field of Hahn series, 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, and be an irreducible element of . Since every principal ideal domain is a unique factorization domain, 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 In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every Ideal (ring_theory)#Examples and properties, nonzero proper ideal factors into a product of prime ideals. It can be shown t ...
. 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 In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of integers. T ...
* Field norm * Absolute value (algebra)


Notes


References

* *. A masterpiece on
algebra Algebra () is one of the areas of mathematics, 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 mathem ...
written by one of the leading contributors. *Chapter VI of *


External links

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