Definition:P-adic Valuation/Integers

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Definition

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

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

$\map {\nu_p^\Z} n := \begin {cases} +\infty & : n = 0 \\ \sup \set {v \in \N: p^v \divides n} & : n \ne 0 \end{cases}$

where:

$\sup$ denotes supremum

$p^v \divides n$ expresses that $p^v$ divides $n$.

Also known as

As the $p$-adic valuation is usually defined as its extension $\nu_p^\Q$ to the rationals, the $p$-adic valuation on $\Z$ is often seen referred to as the restricted $p$-adic valuation.

Also see

• Restricted P-adic Valuation is Valuation

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 2.1$: Absolute Values on a Field: Definition $2.1.2$

• 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.4$ The field of $p$-adic numbers $\Q_p$