Valuation (algebra)


In algebra, 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:
The ordering and group law on are extended to the set by the rules
Then a valuation of is any map
which satisfies the following properties for all a, b in K:
A valuation v is trivial if v = 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. 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 smaller order.
For many applications, is an additive subgroup of the real numbers 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 with the operations of minimum and addition form a semiring, called the min tropical semiring, 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 the same concept writing the group in multiplicative notation as : 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:
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, and v is an absolute value. In this case, we may pass to the additive notation with value group by taking v+ = −log v.
Each valuation on K defines a corresponding linear preorder: a ≼ b ⇔ v ≤ v. Conversely, given a '≼' satisfying the required properties, we can define valuation v =, 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:
There are several objects defined from a given valuation ;

Equivalence of valuations

Two valuations v1 and v2 of with valuation group Γ1 and Γ2, respectively, are said to be equivalent if there is an order-preserving group isomorphism such that v2 = φ 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

Extension of valuations

Let v be a valuation of and let L be a field extension of. An extension of v is a valuation w of L such that the restriction 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 of Γv in Γw, e = , is called the reduced ramification index of w over v. It satisfies e ≤ . The relative degree of w over v is defined to be f = . 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 epi, 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, 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 but stronger in general.

Examples

p-adic valuation

The most basic example is the -adic valuation vp associated to a prime integer p, on the rational numbers with valuation ring The valuation group is the additive integers For an integer the valuation vp measures the divisibility of a by powers of p:
and for a fraction, vp = vp − vp.
Writing this multiplicatively yields the -adic absolute value, which conventionally has as base, so.
The completion of with respect to vp is the field of p-adic numbers.

Order of vanishing

Let K = F, the rational functions on the affine line X = F1, and take a point a ∈ X. For a polynomial with, define va = k, the order of vanishing at x = a; and va = va − va. 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). This can be generalized to the field of Puiseux series K, the Levi-Civita field, 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 uniquely as
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 π-adic valuation of K' is then given by
If π' is another irreducible element of such that = , 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 = .

''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 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 .

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 va on measures the multiplicity of the point x = a in the zero set of f; one may consider this as the order of contact of the graph y = f with the x-axis y = 0 near the point. If, instead of the x-axis, one fixes another irreducible plane curve h = 0 and a point, one may similarly define a valuation vh on so that vh is the order of contact between the fixed curve and f = 0 near. This valuation naturally extends to rational functions
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 vh 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 as:
with the special point corresponding to t = 0. Now define as the order of the formal power series in obtained by restriction of any non-zero polynomial to the curve :
This extends to the field of rational functions by, along with.
Some intersection numbers:

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 is infinite.
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 but if A is radial then f will be radial under the additional condition that f is surjective.