Definition 3

Let $K$ be a (not necessarily commutative) field, $v$ a valuation on $K$ and $\Gamma$ the order group of $v$. $v$ is called discrete if there exists a (necessarily unique) isomorphism of the ordered group $\Gamma$ onto $\mathbf{Z}$. Let $\gamma$ be the element of $\Gamma$ corresponding to $1$ under this isomorphism; every element $u$ of $K$ such that $v(u) = \gamma$ is called a uniformizer of $v$. A discrete valuation is called normed if its order group is $\mathbf{Z}$.

For example the valuation $v_p$ defined by an extremal element $p$ of a principal ideal or factorial, domain is a normed discrete valuation which admits $p$ as a uniformizer. In particular, if $k$ is a field, $k[[T]]$ is the ring of a discrete valuation on $k((T))$ which admits $T$ as a uniformizer. Let $S$ be a connected complex analytic variety of dimension 1, $K$ the field of meromorphic functions on $S$ and $z_0$ a point of $S$; the set off $\in K$ which are holomorphic at $z_0$ is the ring of a discrete valuation $v$; for a function $f \in K$ to be uniformizing for $v$, it is necessary and sufficient that it be holomorphic and zero at $z_0$ and that there exist a neighbourhood $V$ of $z_0$ in $S$ such that the restriction off to $V$ be a homomorphism of $V$ onto a neighbourhood of the origin in $\mathbf{C}$. It is this example and other analogues which are the origin of the word "uniformizer".