A discrete valuation on a field $K$ is a valuation $|\cdot|: K \to \R$ whose image is a discrete subset of $\R$ .

For any field $K$ with a discrete valuation $|\cdot|$ , the set $$ R := \{x \in K : |x| \leq 1\} $$ is a subring of $K$ with sole maximal ideal $$ M := \{x \in K : |x| < 1\}, $$ and hence $R$ is a discrete valuation ring. Conversely, given any discrete valuation ring $R$ , the field of fractions $K$ of $R$ admits a discrete valuation sending each element $x \in R$ to $c^n$ , where $0 < c < 1$ is some arbitrary fixed constant and $n$ is the order of $x$ , and extending multiplicatively to $K$ .

Note: Discrete valuations are often written additively instead of multiplicatively; under this alternate viewpoint, the element $x$ maps to $\log_c|x|$ (in the above notation) instead of just $|x|$ . This transformation reverses the order of the absolute values (since $c < 1$ ), and sends the element $0 \in K$ to $\infty$ . It has the advantage that every valuation can be normalized by a suitable scalar multiple to take values in the integers.