Discrete valuation

Remarks

Example: for a prime $p$, the p-adic valuation $v_p$ on $\Q$ is a discrete valuation. It is a valuation, and $v_p(p^n) = n$ for every $n \in \Z$, so it is onto $\Z$; the prime $p$ is a uniformizer. Example: for a field $k$, the order of vanishing at $0$ on $k(t)$, given by $v(t^n f/g) = n$ for polynomials $f, g$ with $f(0) g(0) \ne 0$, is a discrete valuation with uniformizer $t$. Non-example: the trivial valuation on a field, with $v(x) = 0$ for all $x \ne 0$, takes values in $\Z$ but has $v(K^\times) = \set{0} \ne \Z$. Some texts call a valuation discrete when its value group is isomorphic to $\Z$; dividing by a positive generator then gives a valuation onto $\Z$, called normalized. Every $x \in K^\times$ is $x = u \pi^{v(x)}$ with $v(u) = 0$. The set $\set{x : v(x) \ge 0}$ is a discrete valuation ring with maximal ideal $\pi \cdot \set{x : v(x) \ge 0}$.