Discrete valuation

Examples

• For a fixed prime ${\displaystyle p}$ and for any element ${\displaystyle x\in \mathbb{Q}}$ different from zero write ${\displaystyle x=p^j\frac{a}{b}}$ with ${\displaystyle j,a,b\in \mathbb{Z}}$ such that ${\displaystyle p}$ does not divide ${\displaystyle a,b}$. Then ${\displaystyle \nu (x)=j}$ is a discrete valuation on ${\displaystyle \mathbb{Q}}$, called the p-adic valuation.
• Given a Riemann surface ${\displaystyle X}$, we can consider the field ${\displaystyle K=M(X)}$ of meromorphic functions ${\displaystyle X\to \mathbb{C}\cup \{\mathrm{\infty }\}}$. For a fixed point ${\displaystyle p\in X}$, we define a discrete valuation on ${\displaystyle K}$ as follows: ${\displaystyle \nu (f)=j}$ if and only if ${\displaystyle j}$ is the largest integer such that the function ${\displaystyle f(z)/(z-p{)}^j}$ can be extended to a holomorphic function at ${\displaystyle p}$. This means: if ${\displaystyle \nu (f)=j>0}$ then ${\displaystyle f}$ has a root of order ${\displaystyle j}$ at the point ${\displaystyle p}$; if ${\displaystyle \nu (f)=j<0}$ then ${\displaystyle f}$ has a pole of order ${\displaystyle -j}$ at ${\displaystyle p}$. In a similar manner, one also defines a discrete valuation on the function field of an algebraic curve for every regular point ${\displaystyle p}$ on the curve.

More examples can be found in the article on discrete valuation rings.