Category:P-adic Valuations

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This category contains results about $p$-adic valuations.
Definitions specific to this category can be found in Definitions/P-adic Valuations.

Let $p \in \N$ be a prime number.


The $p$-adic valuation (on $\Z$) is the mapping $\nu_p^\Z: \Z \to \N \cup \left\{{+\infty}\right\}$ defined by:

$\nu_p^\Z \left({n}\right) := \begin{cases} +\infty & : n = 0 \\ \sup \left\{{v \in \N: p^v \mathbin \backslash n}\right\} & : n \ne 0 \end{cases}$


$\sup$ denotes supremum
$p^v \mathbin \backslash n$ expresses that $p^v$ divides $n$.

Rational Numbers

Let the $p$-adic valuation on the integers $\nu_p^\Z$ be extended to $\nu_p^\Q: \Q \to \Z \cup \left\{{+\infty}\right\}$ by:

$\nu_p^\Q \left({\dfrac a b}\right) := \nu_p^\Z \left({a}\right) - \nu_p^\Z \left({b}\right)$

This mapping $\nu_p^\Q$ is called the $p$-adic valuation (on $\Q$) and is usually denoted $\nu_p: \Q \to \Z \cup \left\{{+\infty}\right\}$.


This category has only the following subcategory.