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 the 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. 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 ∪ {∞} 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. 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 larger order. For many applications, Γ 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.