# 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

$$R:=\{x\in K:|x|\le 1\}$$ |

is a subring of $K$ with sole maximal ideal^{}

$$ |

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 $$ 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 ${\mathrm{log}}_{c}|x|$ (in the above notation) instead of just $|x|$. This transformation reverses the order of the absolute values^{} (since $$), and sends the element $0\in K$ to $\mathrm{\infty}$. It has the advantage that every valuation can be normalized by a suitable scalar multiple to take values in the integers.

Title | discrete valuation |
---|---|

Canonical name | DiscreteValuation |

Date of creation | 2013-03-22 13:59:14 |

Last modified on | 2013-03-22 13:59:14 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 6 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 13F30 |

Classification | msc 12J20 |

Synonym | rank one valuations |

Synonym | rank-one valuations |

Related topic | DiscreteValuationRing |

Related topic | Valuation |