There are two different notions of a discrete valuation.
A discrete valuation on a field is a function such that
defines the homomorphism of groups where is the multiplicative group of
with usual conventions for .
The set is called the valuation ring of the valuation , and it is an integral domain with quotient field ; its part is a maximal ideal of and is called the valuation ideal of . This can be generalized to other valuations.
Let and define . Then will be a nonarchimedean multiplicative discrete valuation in the sense of the following definition. If we change a then we get an equivalent multiplicative valuation.
A valuation on a field is a function such that
there is a constant such that if
Two valuations on the same field are equivalent if one is the positive power of another i.e. such that for all .
A valuation is non-archimedean if above can be taken and archimedean otherwise. A (multiplicative) valuation is discrete if there is a neighborhood such that the only such that is .
By a theorem of Gel’fand and Tornheim the only archimedean valuation fields are the subfields of with a valuation which is equivalent to the valuation obtained by resriction from the standard absolute value on .