Valuation (algebra) · Lablypedia

A field with a valuation on it is called a valued field.

Definition

Discrete valuations

A discrete valuation on a field K is a function:

ν : K \to Z \cup {\infty}

satisfying the conditions:

ν (x \cdot y) = ν (x) + ν (y)

ν (x + y) \geq min {ν (x), ν (y)}

ν (x) = \infty ⟺ x = 0

for all $x, y \in K$ .

Note that often the trivial valuation which takes on only the values $0, \infty$ is explicitly excluded.

A field with a non-trivial discrete valuation is called a discrete valuation field.

Relation to discrete valuation rings

To every field $K$ with discrete valuation $ν$ we can associate the subring

O_{ν} := {x \in K ∣ ν (x) \geq 0}

of $K$ , which is a discrete valuation ring. Conversely, the valuation $ν : A \to Z \cup {\infty}$ on a discrete valuation ring $A$ can be extended in a unique way to a discrete valuation on the quotient field $K = Quot (A)$ ; the associated discrete valuation ring $O_{ν}$ is just $A$ .

Discrete valuation rings $O_{ν}$ are local rings with maximal ideal

m_{ν} := {x \in O_{ν} ∣ ν (x) > 0},

so there is a notion of residue field $κ = O_{ν} / m_{ν}$ . Some authors call $O_{ν}$ a local $K$ -algebra when $κ$ and $K$ are isomorphic.

General (Krull) valuations

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 ∪ {∞} 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. 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 and the sum may have larger order.

For many applications, Γ is an additive subgroup of the real numbers $R$ 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, 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.

Associated objects

Similarly to the discrete case, 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 assumed surjective so that Γ_v = Γ),
the valuation ring R_v is the set (subring) of a ∈ K with v(a) ≥ 0,
the prime ideal m_v is the set of a ∈ K with v(a) > 0 (it is in fact a maximal ideal of R_v, and the only such),
the residue field k_v = R_v/m_v.

Multiplicative notation and absolute values

The concept was developed by Emil Artin in his book Geometric Algebra 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:

|a|_v = O if and only if a = 0,
|ab|_v = |a|_v · |b|_v,
|a+b|_v ≤ max(|a|_v, |b|_v), with equality if |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 ≤ |a|_v + |b|_v, and | ⋅ |_v is an absolute value. In this case, we may pass to the additive notation with value group $Γ_{+} \subseteq (R, +)$ by taking v₊(a) = −log |a|_v.

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 ≼.

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".

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Basic properties

Equivalence of valuations and relation to valuation rings

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. On the other hand, any valuation ring O of K is the valuation ring of a canonical valuation v : K → Γ ∪ {∞} , where Γ = K^×/O^×, an abelian group linearly ordered by xO^× ≤ yO^× whenever y/x ∈ O, and v is the quotient map v(x) = xO^×.

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_v] (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 R_w/m_w over 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 this metric to construct its Cauchy completion and obtain a unique valuation extending the one on K, as in the examples below; 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

An elementary example is the p-adic valuation ν_p associated to a prime integer p, on the rational numbers $K = Q,$ with valuation ring $R = Z_{(p)},$ where $Z_{(p)}$ is the localization of $Z$ at the prime ideal $(p)$ . The valuation group is the additive integers $Γ = Z;$ the residue field is $k = Z_{(p)} / p Z_{(p)} ≅ Z / p Z .$ For an integer $a \in R = Z,$ the valuation ν_p(a) measures the divisibility of a by powers of p:

ν_{p} (a) = max {e \in Z ∣ p^{e} divides a};

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 $1 / p = p^{- 1}$ , so $| a |_{p} := p^{- ν_{p} (a)}$ .

The completion of $Q$ with respect to ν_p is the field $Q_{p}$ of p-adic numbers.

The p-adic valuation on $Q_{p}$ extends uniquely to its algebraic closure $\bar{Q_{p}}$ , which has valuation group $Γ = Q$ and the integral closure of $Z_{p}$ as its valuation ring. Hence, $C_{p}$ , the completion of $\bar{Q_{p}}$ (which is again algebraically closed), also has valuation group $Q$ .

Order of vanishing

Given a Riemann surface $X$ , we can consider the field $K = M (X)$ of meromorphic functions $X \to C \cup {\infty}$ . For a fixed point $p \in X$ , we define a discrete valuation on $K$ as follows: $ν (f) = j$ if and only if $j$ is the largest integer such that the function $f (z) / (z - p)^{j}$ can be extended to a holomorphic function at $p$ . This means: if $ν (f) = j > 0$ then $f$ has a root of order $j$ at the point $p$ ; if $ν (f) = j < 0$ then $f$ has a pole of order $- j$ at $p$ . In a similar manner, one also defines a discrete valuation on the function field of an algebraic curve for every regular point $p$ on the curve.

Let K = F(x), the rational functions on the affine line X = F¹, and take a point a ∈ X. For a polynomial $f (x) = a_{k} (x - a)^{k} + a_{k + 1} (x - a)^{k + 1} + \dots + a_{n} (x - a)^{n}$ with $a_{k} \neq 0$ , 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

a = π^{e_{a}} p_{1}^{e_{1}} p_{2}^{e_{2}} \dots p_{n}^{e_{n}}

where the e_i are non-negative integers and the p_i are irreducible elements of R that are not associates of π. In particular, the integer e_a is uniquely determined by a.

The π-adic valuation of K is then given by

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

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). Its valuation ring is R_P.

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_P of R_P yields the P-adic valuation of K.

For instance, if R is the ring of integers of an algebraic number field K, the completion of K with respect to a P-adic valuation is a finite extension of $Q_{p}$ for the prime number p with $P \cap Q = (p)$ , a (non-Archimedean) local field.

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 $f^{- 1} (B)$ . 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.