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

References

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See also