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
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
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
Multiplicity may refer to: In science and the humanities
* Multiplicity (mathematics), the number of times an element is repeated in a multiset
* Multiplicity (philosophy), a philosophical concept
* Multiplicity (psychology), having or using mult ...
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
Contact may refer to:
Interaction Physical interaction
* Contact (geology), a common geological feature
* Contact lens or contact, a lens placed on the eye
* Contact sport, a sport in which players make contact with other players or objects
* ...
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
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
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
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
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
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 referred t ...
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
In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used fo ...
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
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 ...
''R
v'' is the set of ''a'' ∈ with ''v''(''a'') ≥ 0,
*the prime ideal ''m
v'' 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 ''R
v''),
*the residue
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
''k
v'' = ''R
v''/''m
v'',
*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 Γ
1 and Γ
2, respectively, are said to be equivalent if there is an order-preserving
group isomorphism such that ''v''
2(''a'') = φ(''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 ...
these are precisely the equivalence classes of valuations for the
''p''-adic completions of
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'' : Γ''v''">“''w'' : Γ''v'' is called the reduced ramification index of ''w'' over ''v''. It satisfies e(''w''/''v'') ≤
'L'' : ''K''(the
degree of the extension ''L''/''K''). The relative degree of ''w'' over ''v'' is defined to be ''f''(''w''/''v'') =
w''/''mw'' : ''Rv''/''mv''">'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'')''p
i'', where ''p
i'' is the
inseparable degree of the extension ''R
w''/''m
w'' over ''R
v''/''m
v''.
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
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathem ...
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
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
with valuation ring
where
is the localization of
at the prime ideal
. The valuation group is the additive integers
For an integer
the valuation ν
''p''(''a'') measures the divisibility of ''a'' by powers of ''p'':
:
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
, so
.
The
completion of
with respect to ν
''p'' is the field
of
p-adic numbers
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extensio ...
.
Order of vanishing
Let K = F(x), the rational functions on the affine line X = F
1, and take a point ''a'' ∈ X. For a polynomial
with
, 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
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sum ...
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
In mathematics, the Levi-Civita field, named after Tullio Levi-Civita, is a non-Archimedean ordered field; i.e., a system of numbers containing infinite and infinitesimal quantities. Each member a can be constructed as a formal series of the form
...
(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
:
where the ''es are non-negative integers and the ''p
i'' are irreducible elements of that are not
associates of . In particular, the integer ''e
a'' is uniquely determined by ''a''.
The π-adic valuation of ''K'' is then given by
*
*
If π' is another irreducible element of 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).
''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 ''R
P'', is a principal ideal domain whose field of fractions is . The construction of the previous section applied to the prime ideal ''PR
P'' of ''R
P'' 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
. 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)