A discrete valuation ring is a principal ideal domain with exactly one nonzeromaximal ideal. Any generator of is called a uniformizer or uniformizing
element of ; in other words, a uniformizer of is an element such that but
.
Given a discrete valuation ring and a uniformizer , every element can be written uniquely in the form
for some unit and some nonnegative integer. The integer is called the order of , and its value is independent of the choice of uniformizing element .
