P-adic valuation

In number theory, for a given prime number ${\displaystyle p}$, the ${\displaystyle p}$-adic order or ${\displaystyle p}$-adic valuation of a non-zero integer ${\displaystyle n}$ is the highest exponent ${\displaystyle \nu }$ such that ${\displaystyle p^{\nu }}$ divides ${\displaystyle n}$. The ${\displaystyle p}$-adic valuation of ${\displaystyle 0}$ is defined to be ${\displaystyle \infty }$. It is commonly denoted ${\displaystyle \nu _{p}(n)}$. If ${\displaystyle n/d}$ is a rational number in lowest terms, so that ${\displaystyle n}$ and ${\displaystyle d}$ are relatively prime, then ${\displaystyle \nu _{p}(n/d)}$ is equal to ${\displaystyle \nu _{p}(n)}$ if ${\displaystyle p}$ divides ${\displaystyle n}$, or ${\displaystyle -\nu _{p}(d)}$ if ${\displaystyle p}$ divides ${\displaystyle d}$, or to ${\displaystyle 0}$ if it divides neither. The most important application of the ${\displaystyle p}$-adic order is in constructing the field of ${\displaystyle p}$-adic numbers. It is also applied toward various more elementary topics, such as the distinction between singly and doubly even numbers.

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