P-adic valuation

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

The p-adic valuation is a valuation and gives rise to an analogue of the usual absolute value, though unlike the latter, the p-adic absolute value is not Archimedean. Whereas the completion of the rational numbers with respect to the usual absolute value results in the real numbers${\displaystyle \mathbb {R} }$, the completion of the rational numbers with respect to the p-adic absolute value results in the p-adic numbers${\displaystyle \mathbb {Q} _{p}}$.