# Discrete valuation

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

satisfying the conditions

Note that often the trivial valuation which takes on only the values 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 we can associate the subring

of , which is a discrete valuation ring. Conversely, the valuation on a discrete valuation ring can be extended to a valuation on the quotient field giving a discrete valued field , whose associated discrete valuation ring is just .

## Examples

- For a fixed prime for any element different from zero write with such that does not divide , then is a valuation, called the
*p-adic*valuation.

## References

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