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Discrete valuation - Wikipedia
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In mathematics, a discrete valuation is an integer valuation on a field K; that is, a function:
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ν : K → Z ∪ { ∞ } {\displaystyle \nu :K\to \mathbb {Z} \cup \{\infty \}}
satisfying the conditions:
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ν ( x ⋅ y ) = ν ( x ) + ν ( y ) {\displaystyle \nu (x\cdot y)=\nu (x)+\nu (y)}
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ν ( x + y ) ≥ min { ν ( x ) , ν ( y ) } {\displaystyle \nu (x+y)\geq \min {\big \{}\nu (x),\nu (y){\big \}}}
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ν ( x ) = ∞ ⟺ x = 0 {\displaystyle \nu (x)=\infty \iff x=0}
for all x , y ∈ K {\displaystyle x,y\in K} .
Note that often the trivial valuation which takes on only the values 0 , ∞ {\displaystyle 0,\infty } is explicitly excluded.
A field with a non-trivial discrete valuation is called a discrete valuation field.
Discrete valuation rings and valuations on fields[edit]
To every field K {\displaystyle K} with discrete valuation ν {\displaystyle \nu } we can associate the subring
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O K := { x ∈ K ∣ ν ( x ) ≥ 0 } {\displaystyle {\mathcal {O}}_{K}:=\left\{x\in K\mid \nu (x)\geq 0\right\}}
of K {\displaystyle K} , which is a discrete valuation ring. Conversely, the valuation ν : A → Z ∪ { ∞ } {\displaystyle \nu :A\rightarrow \mathbb {Z} \cup \{\infty \}} on a discrete valuation ring A {\displaystyle A} can be extended in a unique way to a discrete valuation on the quotient field K = Quot ( A ) {\displaystyle K={\text{Quot}}(A)} ; the associated discrete valuation ring O K {\displaystyle {\mathcal {O}}_{K}} is just A {\displaystyle A} .
- For a fixed prime p {\displaystyle p} and for any element x ∈ Q {\displaystyle x\in \mathbb {Q} } different from zero write x = p j a b {\displaystyle x=p^{j}{\frac {a}{b}}} with j , a , b ∈ Z {\displaystyle j,a,b\in \mathbb {Z} } such that p {\displaystyle p} does not divide a , b {\displaystyle a,b} . Then ν ( x ) = j {\displaystyle \nu (x)=j} is a discrete valuation on Q {\displaystyle \mathbb {Q} } , called the p-adic valuation.
- Given a Riemann surface X {\displaystyle X} , we can consider the field K = M ( X ) {\displaystyle K=M(X)} of meromorphic functions X → C ∪ { ∞ } {\displaystyle X\to \mathbb {C} \cup \{\infty \}} . For a fixed point p ∈ X {\displaystyle p\in X} , we define a discrete valuation on K {\displaystyle K} as follows: ν ( f ) = j {\displaystyle \nu (f)=j} if and only if j {\displaystyle j} is the largest integer such that the function f ( z ) / ( z − p ) j {\displaystyle f(z)/(z-p)^{j}} can be extended to a holomorphic function at p {\displaystyle p} . This means: if ν ( f ) = j > 0 {\displaystyle \nu (f)=j>0} then f {\displaystyle f} has a root of order j {\displaystyle j} at the point p {\displaystyle p} ; if ν ( f ) = j < 0 {\displaystyle \nu (f)=j<0} then f {\displaystyle f} has a pole of order − j {\displaystyle -j} at p {\displaystyle p} . In a similar manner, one also defines a discrete valuation on the function field of an algebraic curve for every regular point p {\displaystyle p} on the curve.
More examples can be found in the article on discrete valuation rings.
- Cassels, J.W.S.; Fröhlich, Albrecht, eds. (1967), Algebraic Number Theory, Academic Press, Zbl 0153.07403
- Fesenko, Ivan B.; Vostokov, Sergei V. (2002), Local fields and their extensions, Translations of Mathematical Monographs, vol. 121 (Second ed.), Providence, RI: American Mathematical Society, ISBN 978-0-8218-3259-2, MR 1915966
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