Discrete valuation

In mathematics, a discrete valuation is an integer valuation on a field k, that is a function

$\nu:k\to\mathbb Z\cup\{\infty\}$

satisfying the conditions

$\nu(x\cdot y)=\nu(x)+\nu(y)$

$\nu(x+y)\geq\min\big\{\nu(x),\nu(y)\big\}$

$\nu(x)=\infty\iff x=0.$

Note that often the trivial valuation which takes on only the values $0,\infty$ is explicitly excluded.

A field with a non-trivial discrete valuation is called a discrete valuation field.

Discrete valuation rings and valuations on fields

To every field with discrete valuation $\nu$ we can associate the subring

$\mathcal{O}_k := \left\{ x \in k \mid \nu(x) \geq 0 \right\}$

of $k$, which is a discrete valuation ring. Conversely, the valuation $\nu: A \rightarrow \Z\cup\{\infty\}$ on a discrete valuation ring $A$ can be extended to a valuation on the quotient field $\text{Quot}(A)$ giving a discrete valued field $k$, whose associated discrete valuation ring $\mathcal{O}_k$ is just $A$.

Examples

• For a fixed prime $p$ for any element $x \in \mathbb{Q}$ different from zero write $x = p^j\frac{a}{b}$ with $j, a,b \in \Z$ such that $p$ does not divide $a,b$, then $\nu(x) = j$ is a valuation, called the p-adic valuation.

References

Fesenko, Ivan B.; Vostokov, Sergei V. (2002), Local fields and their extensions, Translations of Mathematical Monographs, 121 (Second ed.), Providence, RI: American Mathematical Society, ISBN 978-0-821-83259-2, MR1915966 

See also

• discrete valuation ring
• Valuation (algebra)
• Valuation ring