There are two different notions of a discrete valuation.
A discrete valuation on a field $K$ is a function $v:K\to Z\cup \{\mathrm{\infty}\}$ such that
$v$ defines the homomorphism of groups $v{\mid}_{K}:{K}^{*}\to Z$ where ${K}^{*}$ is the multiplicative group of $K$
$v(0)=\mathrm{\infty}$
$v(x+y)\ge \mathrm{inf}\{v(x),v(y)\}$
with usual conventions for $\mathrm{\infty}$.
The set ${R}_{v}=\{x\in K\phantom{\rule{thinmathspace}{0ex}}\mid \phantom{\rule{thinmathspace}{0ex}}v(x)\ge 0\}$ is called the valuation ring of the valuation $v$, and it is an integral domain with quotient field $K$; its part ${\U0001d52d}_{v}=\{x\in K\phantom{\rule{thinmathspace}{0ex}}\mid \phantom{\rule{thinmathspace}{0ex}}v(x)>0\}$ is a maximal ideal of ${R}_{v}$ and is called the valuation ideal of $v$. This can be generalized to other valuations.
Let $0<\rho <1$ and define $\mid x{\mid}_{v}={\rho}^{v(x)}$. Then $\mid \mid $ will be a nonarchimedean multiplicative discrete valuation in the sense of the following definition. If we change a $\rho $ then we get an equivalent multiplicative valuation.
A valuation on a field $K$ is a function $\mid \mid :K\to {R}_{\ge 0}$ such that
$\mid x\mid =0$ iff $x=0$
$\mid xy\mid =\mid x\mid \mid y\mid $
there is a constant $C$ such that $\mid 1+x\mid \le C$ if $\mid x\mid \le 1$
Two valuations on the same field are equivalent if one is the positive power of another i.e. $\exists c>0$ such that $\mid x{\mid}_{1}=\mid x{\mid}_{2}^{c}$ for all $x\in K$.
A valuation is non-archimedean if $C$ above can be taken $1$ and archimedean otherwise. A (multiplicative) valuation is discrete if there is a neighborhood $U\ni 1\in {R}_{+}$ such that the only $x\in K$ such that $\mid x\mid \in U$ is ${1}_{K}\in K$.
By a theorem of Gel’fand and Tornheim the only archimedean valuation fields are the subfields of $C$ with a valuation which is equivalent to the valuation obtained by resriction from the standard absolute value on $C$.