Discrete valuation

Summary

A discrete valuation is a valuation on a field whose values on nonzero elements are exactly the integers, see the definition. It counts the power of a fixed "prime" element in each nonzero element. The p-adic valuation on $\Q$ and the order of vanishing of a rational function at a point are discrete valuations. The elements of nonnegative value form a discrete valuation ring, one of the basic rings of number theory and algebraic geometry.