P-adic valuation

Definition

Let $p$ be a prime number and let $n$ be a nonzero integer. The $p$ -adic valuation $\nu_p(n)$ is the largest natural number $k$ such that $p^k \mid n$ . By convention, $\nu_p(0) = \infty$ .

Remarks

For a positive integer $n$ , $\nu_p(n)$ is the exponent of $p$ in the prime factorization of $n$ . It is $0$ when $p \nmid n$ . For example, $\nu_2(40) = 3$ and $\nu_5(40) = 1$ , because $40 = 2^3 \cdot 5$ . For a nonzero rational number $a / b$ , one sets $\nu_p(a / b) = \nu_p(a) - \nu_p(b)$ . The Lean library Mathlib sets $\nu_p(0) = 0$ for its function on natural numbers.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Divisibility definition
Prime number definition
P-adic valuation definition

Where this leads

5 pages build on this one.

Tag	`04VCPS30ZE`
URI	`m2://04VCPS30ZE`
Aliases	`p-adic-order`, `multiplicity-of-a-prime`
Kind	definition
Writing	W1 stated
MSC	11A51
Defines	P-adic valuation
Uses	Prime number, Divisibility

P-adic valuation

Definition

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

What links here

Used by