For a fixed and for any element different from zero write with such that does not divide . Then is a discrete valuation on , called the p-adic valuation.
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Given a , we can consider the field of meromorphic functions. For a fixed point , we define a discrete valuation on as follows: if and only if is the largest integer such that the function can be extended to a holomorphic function at . This means: if then has a root of order at the point ; if then has a pole of order at . In a similar manner, one also defines a discrete valuation on the function field of an algebraic curve for every regular point on the curve.