In algebra (in particular in algebraic geometry or algebraic number theory), a **valuation** is a function on a field that provides a measure of the 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**.

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 ∪ {∞}^{[a]} 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 ∪ {∞}

that 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 on *K*^{×}. 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.^{[b]} The third property then corresponds to the order of a sum being the order of the larger term,^{[c]} unless the two terms have the same order, in which case they may cancel and the sum may have larger order.

For many applications, Γ is an additive subgroup of the real numbers ^{[d]} 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,^{[e]} 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.

The concept was developed by Emil Artin in his book *Geometric Algebra* writing the group in multiplicative notation as (Γ, ·, ≥):^{[1]}

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*:

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

(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 *|a+b| _{v}* ≤

Each valuation on *K* defines a corresponding linear preorder: *a* ≼ *b* ⇔ *|a| _{v}* ≤

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".

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.

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 these are precisely the equivalence classes of valuations for the *p*-adic completions of

Let *v* be a valuation of K and let *L* be a field extension of K. An **extension of v** (to

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}*/

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 but stronger in general.

The most basic example is the p-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 p-adic absolute value, which conventionally has as base , so .

The completion of with respect to ν_{p} is the field of p-adic numbers.

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.

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

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).

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

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

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.

**^**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.

**^**Emil Artin*Geometric Algebra*, pages 47 to 49, via Internet Archive

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*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, vol. 29, New York, Heidelberg: Springer-Verlag, ISBN 978-0-387-90171-8, Zbl 0322.13001 - Schaefer, Helmut H.; Wolff, M.P. (1999).
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