Uniformising parameter

In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal.

This means a DVR is an integral domain R which satisfies any one of the following equivalent conditions:

Given an algebraic curve ${\displaystyle (X,{\mathcal {O}}_{X})}$ the local ring at a smooth point ${\displaystyle {\mathcal {O}}_{\mathfrak {p}}}$ is a discrete valuation ring because it is a principal valuation ring.

Let Z₍₂₎ := { z ⁄ n : z, n ∈ Z, n odd }. Then the field of fractions of Z₍₂₎ is Q. Now, for any nonzero element r of Q, we can apply unique factorization to the numerator and denominator of r to write r as 2^kz ⁄ n, where z, n, and k are integers with z and n odd. In this case, we define ν(r)=k. Then Z₍₂₎ is the discrete valuation ring corresponding to ν. The maximal ideal of Z₍₂₎ is the principal ideal generated by 2, and the "unique" irreducible element (up to units) is 2.

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