Discrete valuation

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In mathematics, a discrete valuation on a field k 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\mathrm{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.

[edit] Discrete Valuation Rings and valuations on fields

To every field with discrete valuation ν 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. Contrarily, the valuation \nu: A \rightarrow \Z\cup\{\infty\} on a discrete valuation ring A can be extended to a valuation on the quotient field Quot(A) giving a discrete valued field k, whose associated discrete valuation ring \mathcal{O}_k is just A.

[edit] 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 define ν(x): = j.

[edit] See also

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