In mathematics, a discrete valuation is an integer valuation on a field K; that is, a function
ν : K → Z ∪ { ∞ } satisfying the conditions
ν ( x ⋅ y ) = ν ( x ) + ν ( y ) ν ( x + y ) ≥ min { ν ( x ) , ν ( y ) } ν ( x ) = ∞ ⟺ x = 0 for all x , y ∈ K .
Note that often the trivial valuation which takes on only the values 0 , ∞ is explicitly excluded.
A field with a non-trivial discrete valuation is called a discrete valuation field.
Discrete valuation rings and valuations on fields
To every field K with discrete valuation ν we can associate the subring
of K , which is a discrete valuation ring. Conversely, the valuation ν : A → Z ∪ { ∞ } 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 K is just A .
For a fixed prime p and for any element x ∈ Q different from zero write x = p j a b with j , a , b ∈ Z such that p does not divide a , b , then ν ( x ) = j is a discrete valuation on Q , called the p-adic valuation.Given a Riemann surface X , we can consider the field K = M ( X ) of meromorphic functions X → C ∪ { ∞ } . For a fixed point p ∈ 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.More examples can be found in the article on discrete valuation rings.
