Valuation (algebra)

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|_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 ${\displaystyle \Gamma _{+}\subset (\mathbb {R} ,+)}$ 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 ≼.

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_v is the set 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),
• 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 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 ${\displaystyle \mathbb {Q}$ :} these are precisely the equivalence classes of valuations for the p-adic completions of ${\displaystyle \mathbb {Q} .}$

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.

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 ${\displaystyle \Gamma =\mathbb {Z} ,}$ 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 ${\displaystyle K=\mathbb {Q} ,}$ with valuation ring ${\displaystyle R=\mathbb {Z} _{(p)},}$ where ${\displaystyle \mathbb {Z} _{(p)}}$ is the localization of ${\displaystyle \mathbb {Z} }$ at the prime ideal ${\displaystyle (p)}$. The valuation group is the additive integers ${\displaystyle \Gamma =\mathbb {Z} .}$ For an integer ${\displaystyle a\in R=\mathbb {Z} ,}$ the valuation ν_p(a) measures the divisibility of a by powers of p:

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

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

The completion of ${\displaystyle \mathbb {Q} }$ with respect to ν_p is the field ${\displaystyle \mathbb {Q} _{p}}$ of p-adic numbers.

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

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

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

where the e's are non-negative integers and the 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

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

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).

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.