Extension of a valuation
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. They generalize 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.
To define the algebraic concept of valuation, the following objects are needed:
The ordering and group law on Γ are extended to the set Γ ∪ {∞} by the rules
Then a valuation of K is any map
which satisfies the following properties for all a, b in K:
Some authors use the term exponential valuation rather than "valuation". In this case the term "valuation" means "absolute value".
A valuation v is called trivial (or the trivial valuation of K) if v(a) = 0 for all a in K×, otherwise it is called non-trivial.
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 second property asserts that any valuation is a group homomorphism, while the third property is a translation of the triangle inequality from metric spaces to ordered groups.
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