Editing Discrete valuation ring (section)

==<span id="uniformizer"></span>Uniformizing parameter==
Given a DVR ''R'', any irreducible element of ''R'' is a generator for the unique maximal ideal of ''R'' and vice versa. Such an element is also called a '''uniformizing parameter''' of ''R'' (or a '''uniformizing element''', a '''uniformizer''', or a '''prime element''').

If we fix a uniformizing parameter ''t'', then ''M''=(''t'') is the unique maximal ideal of ''R'', and every other non-zero ideal is a power of ''M'', i.e. has the form (''t''<sup>&nbsp;''k''</sup>) for some ''k''≥0. All the powers of ''t'' are distinct, and so are the powers of ''M''. Every non-zero element ''x'' of ''R'' can be written in the form α''t''<sup>&nbsp;''k''</sup> with α a unit in ''R'' and ''k''≥0, both uniquely determined by ''x''. The valuation is given by ''ν''(''x'') = ''kv''(''t''). So to understand the ring completely, one needs to know the group of units of ''R'' and how the units interact additively with the powers of ''t''.

The function ''v'' also makes any discrete valuation ring into a [[Euclidean domain]].{{Citation needed|date=May 2015}}