Discrete valuation
In mathematics, a discrete valuation is an integer valuation on a field K; that is, a function:Template:Sfn
- <math>\nu:K\to\mathbb Z\cup\{\infty\}</math>
satisfying the conditions:
- <math>\nu(x\cdot y)=\nu(x)+\nu(y)</math>
- <math>\nu(x+y)\geq\min\big\{\nu(x),\nu(y)\big\}</math>
- <math>\nu(x)=\infty\iff x=0</math>
for all <math>x,y\in K</math>.
Note that often the trivial valuation which takes on only the values <math>0,\infty</math> is explicitly excluded.
A field with a non-trivial discrete valuation is called a discrete valuation field.
Discrete valuation rings and valuations on fieldsEdit
To every field <math>K</math> with discrete valuation <math>\nu</math> we can associate the subring
- <math>\mathcal{O}_K := \left\{ x \in K \mid \nu(x) \geq 0 \right\}</math>
of <math>K</math>, which is a discrete valuation ring. Conversely, the valuation <math>\nu: A \rightarrow \Z\cup\{\infty\}</math> on a discrete valuation ring <math>A</math> can be extended in a unique way to a discrete valuation on the quotient field <math>K=\text{Quot}(A)</math>; the associated discrete valuation ring <math>\mathcal{O}_K</math> is just <math>A</math>.
ExamplesEdit
- For a fixed prime <math>p</math> and for any element <math>x \in \mathbb{Q}</math> different from zero write <math>x = p^j\frac{a}{b}</math> with <math>j, a,b \in \Z</math> such that <math>p</math> does not divide <math>a,b</math>. Then <math>\nu(x) = j</math> is a discrete valuation on <math>\Q</math>, called the p-adic valuation.
- Given a Riemann surface <math>X</math>, we can consider the field <math>K=M(X)</math> of meromorphic functions <math>X\to\Complex\cup\{\infin\}</math>. For a fixed point <math>p\in X</math>, we define a discrete valuation on <math>K</math> as follows: <math>\nu(f)=j</math> if and only if <math>j</math> is the largest integer such that the function <math>f(z)/(z-p)^j</math> can be extended to a holomorphic function at <math>p</math>. This means: if <math>\nu(f)=j>0</math> then <math>f</math> has a root of order <math>j</math> at the point <math>p</math>; if <math>\nu(f)=j<0</math> then <math>f</math> has a pole of order <math>-j</math> at <math>p</math>. In a similar manner, one also defines a discrete valuation on the function field of an algebraic curve for every regular point <math>p</math> on the curve.
More examples can be found in the article on discrete valuation rings.