# 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:

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

• ∞ ≥ α for all αΓ,
• ∞ + α = α + ∞ = ∞ for all αΓ.

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.[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, in which case the sum may have smaller order.

For many applications, Γ is an additive subgroup of the real numbers R,[d] in which case ∞ can be interpreted as +∞ in the extended real numbers; note that $\min(a,+\infty )=\min(+\infty ,a)=a$  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.

### Multiplicative notation and absolute values

We could define 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

• Oα for all αΓ,
• O · α = α · O = O for all αΓ.

Then a valuation of K is any map

v : K → Γ ∪ {O}

satisfying the following properties for all a, bK:

• 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: abv(a) ≤ v(b). Conversely, given a '≼' satisfying the required properties, we can define valuation v(a) = {b: baab}, with multiplication and ordering based on K and ≼.

### Terminology

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 Γv = v(K×), a subgroup of Γ (though v is usually surjective so that Γv = Γ);
• the valuation ring Rv is the set of aK with v(a) ≥ 0,
• the prime ideal mv is the set of aK with v(a) > 0 (it is in fact a maximal ideal of Rv),
• the residue field kv = Rv/mv,
• the place of K associated to v, the class of v under the equivalence defined below.

## Basic properties

### Equivalence of valuations

Two valuations v1 and v2 of K with valuation group Γ1 and Γ2, respectively, are said to be equivalent if there is an order-preserving group isomorphism φ : Γ1 → Γ2 such that v2(a) = φ(v1(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 Γv in Γw, e(w/v) = [Γw : Γv], 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) = [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)pi, where pi is the inseparable degree of the extension Rw/mw over Rv/mv.

### 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

The most basic example is the p-adic valuation vp 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 vp(a) measures the divisibility of a by powers of p:

$v_{p}(a)=\max\{e\in \mathbb {Z} \mid p^{e}{\text{ divides }}a\};$

and for a fraction, vp(a/b) = vp(a) − vp(b).

Writing this multiplicatively yields the p-adic absolute value, which conventionally has as base $1/p=p^{-1}$ , so $|a|_{p}:=p^{-v_{p}(a)}$ .

The completion of Q with respect to vp is the field Qp of p-adic numbers.

### Order of vanishing

Let K = F(x), the rational functions on the affine line X = F1, and take a point a ∈ X. For a polynomial $f(x)=a_{k}(x{-}a)^{k}+a_{k+1}(x{-}a)^{k+1}+\cdots +a_{n}(x{-}a)^{n}$  with $a_{k}\neq 0$ , define va(f) = k, the order of vanishing at x = a; and va(f /g) = va(f) − va(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((xa)). 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

$a=\pi ^{e_{a}}p_{1}^{e_{1}}p_{2}^{e_{2}}\cdots p_{n}^{e_{n}}$

where the e's are non-negative integers and the pi are irreducible elements of R that are not associates of π. In particular, the integer ea is uniquely determined by a.

The π-adic valuation of K is then given by

• $v_{\pi }(0)=\infty$
• $v_{\pi }(a/b)=e_{a}-e_{b},{\text{ for }}a,b\in R,a,b\neq 0.$

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

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.

### 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(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 vh on R = C[x,y] so that vh(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 /gK = 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 $R=\mathbb {C} [x,y]_{(h)}=\{{\tfrac {f}{g}}\in \mathbb {C} (x,y)\mid h{\text{ does not divide }}g\}$ , 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 $(h)^{k}$ . Then the above valuation vh is the π-adic valuation corresponding to the irreducible element π = hR.

Example: Consider the curve $V_{h}$  defined by $h(x,y)=x^{3}-xy-y=0$ , namely the graph $\textstyle y={\frac {x^{3}}{1-x}}=\sum _{n=3}^{\infty }x^{n},$  near the origin $(a,b)=(0,0)$ . This curve can be parametrized by tC as:

$\textstyle (x,y)\!=\!(t,{\frac {t^{3}}{1-t}})=(t,\sum _{n=3}^{\infty }t^{n}),$

with the special point (0,0) corresponding to t = 0. Now define $v_{h}:\mathbf {C} [x,y]\to \mathbf {Z}$  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 Vh:

$\textstyle v_{h}(f)=\mathrm {ord} _{t}(f|_{V_{h}})=\mathrm {ord} _{t}\!\left(f(t,\sum _{n=3}^{\infty }t^{n})\right),\quad {\text{for }}f\in \mathbf {C} [x,y].$

This extends to the field of rational functions C(x, y) by $v_{h}(f/g)=v_{h}(f)-v_{h}(g)$ , along with $v_{h}(0)=\infty$ .

Some intersection numbers:

{\begin{aligned}v_{h}(x)&=\mathrm {ord} _{t}(t)=1\\v_{h}(x^{6}-y^{2})&=\mathrm {ord} _{t}\left(t^{6}-t^{6}-2t^{7}-3t^{8}-\cdots \right)=\mathrm {ord} _{t}\left(-2t^{7}-3t^{8}-\cdots \right)=7\\v_{h}\!\left({\tfrac {x^{6}-y^{2}}{x}}\right)&=\mathrm {ord} _{t}\left(-2t^{7}-3t^{8}-\cdots \right)-\mathrm {ord} _{t}(t)\\&=7-1=6\end{aligned}}

## 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 αK such that λK 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 $f^{-1}(B)$ . 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.