Definition:P-adic Valuation/Rational Numbers

Definition

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

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

$\map {\nu_p^\Q} {\dfrac a b} := \map {\nu_p^\Z} a - \map {\nu_p^\Z} b$

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

Also see

• P-adic Valuation of Rational Number is Well Defined
• P-adic Valuation is Valuation, showing that indeed $\nu_p^\Q$ is a 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 $\Q_p$