Uniformizing element

An element $\pi$ of a discrete valuation ring $A$ (cf. Discretely-normed ring) with prime ideal $\mathfrak p$ such that $\mathfrak p=A\pi$. If $\pi_1,\pi_2$ are two uniformizing elements in $A$, then the element $\pi_1\pi_2^{-1}$ is invertible in $A$. Let $R$ be a system of representatives in $A$ for the elements of the residue field $A/\mathfrak p$. Then any element $a\in A$ can be uniquely expressed as a power series $\sum_{i=0}^\infty r_i\pi^i$, where $r_i\in R$ and $\pi$ is a uniformizing element. If the ring $A$ is complete relative to the discrete valuation, then any power series of the above form represents an element $a\in A$.

If $A$ is the local ring of functions at a simple point $x$ of an algebraic curve $X$, then $\pi$ is a uniformizing element if and only if $\pi$ has a zero of order one at $x$.

References

[a1]	E. Weiss, "Algebraic number theory" , McGraw-Hill (1963) pp. Sects. 4–9