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Homediscrete valuation

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# discrete valuation

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

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.

## Mathematics Subject Classification

13F30*no label found*12J20

*no label found*

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