Valuation (algebra)

For the use of the term in geometry, see Valuation (geometry).

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. In all of these examples, the valuation assumes integer values and is therefore called a discrete valuation, but in general, the integers are replaced by an abelian totally ordered group.

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

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