uniformizer

A uniformizer is a concept from valuation theory, used to describe a single element that generates the maximal ideal of a discrete valuation ring (DVR). If O is a DVR with field of fractions K and maximal ideal m, then an element π ∈ O is a uniformizer precisely when m = πO. Equivalently, there is a valuation v: K× → Z with v(π) = 1, and every nonzero x ∈ K can be written uniquely as x = πn u for some integer n and a unit u in O. Any unit multiple of a uniformizer is itself a uniformizer.

Examples include p in the ring of p-adic integers Zp, and t in the formal power series ring k[[t]] over a field k, where the maximal ideal is (t). In this context, k[[t]] is a DVR and t serves as a uniformizer. In the Laurent series field k((t)), the valuation ring is k[[t]], with t as a uniformizer for the corresponding local structure.

Uniformizers are used to measure orders of vanishing or poles. Writing x as x = πn u with u a unit, the exponent n records the order of vanishing (if n > 0) or the order of the pole (if n < 0) at the corresponding point. In ramification theory, if L/K is a finite extension and π is a uniformizer of K, then π often maps to a uniformizer in L raised to the ramification index, up to a unit.

In algebraic geometry, at a nonsingular point of a curve, a choice of uniformizing parameter t in the local ring OX,x provides a local coordinate. This makes OX,x into a DVR and permits local expansions of functions in powers of t, with the vanishing order given by the exponent of t.