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.
One starts with the following objects: The ordering and group law on Γ are extended to the set Γ ∪ {∞}[a] by the rules Then a valuation of K is any map that satisfies the following properties for all a, b in K: 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.
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 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 Then a valuation of K is any map satisfying the following properties for all a, b ∈ K: (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 ≤ |a|v + |b|v, and | ⋅ |v is an absolute value.
In this case, we may pass to the additive notation with value group
Each valuation on K defines a corresponding linear preorder: a ≼ b ⇔ |a|v ≤ |b|v.
Conversely, given a "≼" satisfying the required properties, we can define valuation |a|v = {b: b ≼ a ∧ a ≼ b}, with multiplication and ordering based on K and ≼.
In this article, we use the terms defined above, in the additive notation.
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
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.
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.
The most basic example is the p-adic valuation νp associated to a prime integer p, on the rational numbers
Writing this multiplicatively yields the p-adic absolute value, which conventionally has as base
Let K = F(x), the rational functions on the affine line X = F1, and take a point a ∈ X.
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 pi are irreducible elements of R that are not associates of 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 RP, is a principal ideal domain whose field of fractions is K. The construction of the previous section applied to the prime ideal PRP of RP yields the P-adic valuation of K. Suppose that Γ ∪ {0} is the set of non-negative real numbers under multiplication.
Radial subsets of X are invariant under finite intersection.
The set of circled subsets of L is invariant under arbitrary intersections.
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.