# Valuation (algebra)

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. It generalizes 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

One starts with the following objects:

- a field K and its multiplicative group
*K*^{×}, - an abelian totally ordered group (Γ, +, ≥).

The ordering and group law on Γ are extended to the set Γ ∪ {∞}[lower-alpha 1] by the rules

- ∞ ≥
*α*for all α ∈ Γ, - ∞ +
*α*=*α*+ ∞ = ∞ for all α ∈ Γ.

Then a **valuation of K** is any map

*v*:*K*→ Γ ∪ {∞}

which satisfies the following properties for all *a*, *b* in *K*:

*v*(*a*) = ∞ if and only if*a*= 0,*v*(*ab*) =*v*(*a*) +*v*(*b*),*v*(*a*+*b*) ≥ min(*v*(*a*),*v*(*b*)), 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 on metric spaces 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.[lower-alpha 2] The third property then corresponds to the order of a sum being the order of the larger term,[lower-alpha 3] unless the two terms have the same order, in which case they may cancel, in which case the sum may have smaller order.

For many applications, Γ is an additive subgroup of the real numbers **R**,[lower-alpha 4] in which case ∞ can be interpreted as +∞ in the extended real numbers; 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, called the min tropical semiring,[lower-alpha 5] 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

We could define[1] the same concept writing the group in multiplicative notation as (Γ, ·, ≥): instead of ∞, we adjoin a formal symbol *O* to Γ, with the ordering and group law extended by the rules

*O*≤*α*for all α ∈ Γ,*O*·*α*=*α*·*O*=*O*for all α ∈ Γ.

Then a **valuation of K** is any map

*v*:*K*→ Γ ∪ {*O*}

satisfying the following properties for all *a*, *b* ∈ *K*:

*v*(*a*) =*O*if and only if*a*= 0,*v*(*ab*) =*v*(*a*) ·*v*(*b*),*v*(*a*+*b*) ≤ max(*v*(*a*),*v*(*b*)), with equality if*v*(*a*) ≠*v*(*b*).

(Note that the directions of the inequalities are reversed from those in the additive notation.)

If Γ is a subgroup of the positive real numbers under multiplication, the last condition is the ultrametric inequality, a stronger form of the triangle inequality *v*(*a* + *b*) ≤ *v*(*a*) + *v*(*b*), and *v* is an absolute value. In this case, we may pass to the additive notation with value group Γ_{+} ⊂ (**R**, +) by taking *v*_{+}(*a*) = −log *v*(*a*).

Each valuation on *K* defines a corresponding linear preorder: *a* ≼ *b* ⇔ *v*(*a*) ≤ *v*(*b*). Conversely, given a '≼' satisfying the required properties, we can define valuation *v*(*a*) = {*b*: *b* ≼ *a* ∧ *a* ≼ *b*}, with multiplication and ordering based on *K* 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 *v* : *K* → Γ ∪ {∞} ;

- the
**value group**or**valuation group**Γ_{v}=*v*(*K*^{×}), a subgroup of Γ (though*v*is usually surjective so that Γ_{v}= Γ); - the
**valuation ring***R*is the set of_{v}*a*∈ K with*v*(*a*) ≥ 0, - the
**prime ideal***m*is the set of_{v}*a*∈*K*with*v*(*a*) > 0 (it is in fact a maximal ideal of*R*),_{v} - the
**residue field***k*=_{v}*R*/_{v}*m*,_{v} - the
**place**of K 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 K with valuation group Γ_{1} and Γ_{2}, respectively, are said to be **equivalent** if there is an order-preserving group isomorphism *φ* : Γ_{1} → Γ_{2} 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* 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 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 restriction of

*w*to K 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 of Γ_{v} in Γ_{w}, e(*w*/*v*) = [Γ_{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*) = [*R _{w}*/

*m*:

_{w}*R*/

_{v}*m*] (the degree of the extension of residue fields). It is also less than or equal to the degree of

_{v}*L*/

*K*. When

*L*/

*K*is separable, the

**ramification index**of

*w*over

*v*is defined to be e(

*w*/

*v*)

*p*, where

^{i}*p*is the inseparable degree of the extension

^{i}*R*/

_{w}*m*over

_{w}*R*/

_{v}*m*.

_{v}### Complete valued fields

When the ordered abelian group Γ is the additive group of the integers, the associated valuation is equivalent to an absolute value, and hence induces a metric on the field K. If K 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 K, and K 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 Γ = **Z**, but stronger in general.

## Examples

### p-adic valuation

The most basic example is the p-adic valuation *v*_{p} associated to a prime integer *p*, on the rational numbers *K* = **Q**, with valuation ring *R* = **Z**. The valuation group is the additive integers Γ = **Z**. For an integer *a* ∈ R = **Z**, the valuation *v*_{p}(*a*) measures the divisibility of *a* by powers of *p*:

and for a fraction, *v*_{p}(*a*/*b*) = *v*_{p}(*a*) − *v*_{p}(*b*).

Writing this multiplicatively yields the p-adic absolute value, which conventionally has as base , so .

The completion of **Q** with respect to *v*_{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** = **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 ring **F**((*x*−*a*)). This can be generalized to the field of Puiseux series *K*{{*t*}} (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.

### π-adic valuation

Generalizing the previous examples, 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

where the *e'*s are non-negative integers and the *p _{i}* are irreducible elements of R that are not associates of π. In particular, the integer

*e*is uniquely determined by

_{a}*a*.

The **π-adic valuation of K** is then given by

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).

*P*-adic valuation on a Dedekind domain

*P*-adic valuation on a Dedekind domain

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 R at *P*, denoted *R _{P}*, is a principal ideal domain whose field of fractions is K. The construction of the previous section applied to the prime ideal

*PR*of

_{P}*R*yields the

_{P}**P-adic valuation of K**.

### Geometric notion of contact

Valuations can be defined for a field of functions on a space of dimension greater than one. Recall that the order-of-vanishing valuation *v*_{a}(*f*) on *R* = **C**[x] measures the multiplicity of the point *x* = *a* in the zero set of *f*; one may consider this as the order of contact (or local intersection number) of the graph *y* = *f*(*x*) with the *x*-axis *y* = 0 near the point (*a*,0). If, instead of the *x*-axis, one fixes another irreducible plane curve *h*(*x*,*y*) = 0 and a point (*a*,*b*), one may similarly define a valuation *v*_{h} on *R* = **C**[*x*,*y*] so that *v*_{h}(*f*) is the order of contact (the intersection number) between the fixed curve and *f*(*x*,*y*) = 0 near (*a*,*b*). This valuation naturally extends to rational functions *f* /*g* ∈ *K* = **C**(*x*,*y*).

In fact, this construction is a special case of the π-adic valuation on a PID defined above. Namely, consider the local ring , the ring of rational functions which are defined on some open subset of the curve *h* = 0. This is a PID; in fact a discrete valuation ring whose only ideals are the powers . Then the above valuation *v*_{h} is the π-adic valuation corresponding to the irreducible element π = *h* ∈ *R*.

Example: Consider the curve defined by , namely the graph near the origin . This curve can be parametrized by *t* ∈ **C** as:

with the special point (0,0) corresponding to *t* = 0. Now define as the order of the formal power series in t obtained by restriction of any non-zero polynomial f in **C**[*x*, *y*] to the curve V_{h}:

This extends to the field of rational functions **C**(*x*, *y*) by , along with .

Some intersection numbers:

## Vector spaces over valuation fields

Suppose that Γ ∪ {0} 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.

## Notes

- The symbol ∞ denotes an element not in Γ, with no other meaning. Its properties are simply defined by the given axioms.
- With the min convention here, the valuation is rather interpreted as the
*negative*of the order of the leading order term, but with the max convention it can be interpreted as the order. - Again, swapped since using minimum convention.
- Every Archimedean group is isomorphic to a subgroup of the real numbers under addition, but non-Archimedean ordered groups exist, such as the additive group of a non-Archimedean ordered field.
- In the tropical semiring, minimum and addition of real numbers are considered
*tropical addition*and*tropical multiplication*; these are the semiring operations.

## References

- Emil Artin (1957) Geometric Algebra, page 48

- Efrat, Ido (2006),
*Valuations, orderings, and Milnor*K*-theory*, Mathematical Surveys and Monographs,**124**, Providence, RI: American Mathematical Society, ISBN 0-8218-4041-X, Zbl 1103.12002 - Jacobson, Nathan (1989) [1980], "Valuations: paragraph 6 of chapter 9",
*Basic algebra II*(2nd ed.), New York: W. H. Freeman and Company, ISBN 0-7167-1933-9, Zbl 0694.16001. A masterpiece on algebra written by one of the leading contributors. - Chapter VI of Zariski, Oscar; Samuel, Pierre (1976) [1960],
*Commutative algebra, Volume II*, Graduate Texts in Mathematics,**29**, New York, Heidelberg: Springer-Verlag, ISBN 978-0-387-90171-8, Zbl 0322.13001 - Schaefer, Helmuth H.; Wolff, M.P. (1999).
*Topological Vector Spaces*. GTM.**3**. New York: Springer-Verlag. pp. 10–11. ISBN 9780387987262.

## External links

- Danilov, V.I. (2001) [1994], "Valuation", in Hazewinkel, Michiel (ed.),
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - Discrete valuation at PlanetMath.org.
- Valuation at PlanetMath.org.
- Weisstein, Eric W. "Valuation".
*MathWorld*.