$p$ -adic valuation

In number theory, the $p$ -adic valuation or $p$ -adic order of an integer $n$ is the exponent of the highest power of the prime number $p$ that divides $n$ . It is denoted $\nu _{p}(n)$ . Equivalently, $\nu _{p}(n)$ is the exponent to which $p$ appears in the prime factorization of $n$ .

The $p$ -adic valuation is a valuation and gives rise to an analogue of the usual absolute value. Whereas the completion of the rational numbers with respect to the usual absolute value results in the real numbers $\mathbb {R}$ , the completion of the rational numbers with respect to the $p$ -adic absolute value results in the $p$ -adic numbers $\mathbb {Q} _{p}$ .^[1]

$p$ -adic number

$p$ -adic valuation

$p$ -adic number

Valuation (algebra)

Archimedean property

Multiplicity (mathematics)

Ostrowski's theorem

Legendre's formula

p-adic valuation

p-adic number

Valuation (algebra)

Archimedean property

Multiplicity (mathematics)

Ostrowski's theorem

Legendre's formula

$p$ -adic valuation

$p$ -adic number