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Math 2.0

Number theory Proposition

P-adic valuation of a product

Builds on P-adic valuation

Statement

Let $p$ be a prime number, and let $a$ and $b$ be nonzero integers. Then $\nu_p(a b) = \nu_p(a) + \nu_p(b)$ , where $\nu_p$ is the p-adic valuation.

Remarks

With the convention $\nu_p(0) = \infty$ , the formula holds for all integers $a$ and $b$ . The proof uses Euclid's lemma. Write $a = p^i a'$ and $b = p^j b'$ with $p \nmid a'$ and $p \nmid b'$ . Then $p \nmid a' b'$ , so $\nu_p(a b) = i + j$ . For example, $\nu_2(12 \cdot 10) = \nu_2(120) = 3 = 2 + 1$ .

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Divisibility definition
Prime number definition
P-adic valuation definition
P-adic valuation of a product statement

Further reading

Web https://en.wikipedia.org/wiki/P-adic_valuation, Definition and properties

About this page

Tag	`0850JJMJYZ`
URI	`m2://0850JJMJYZ`
Kind	statement, proposition
Writing	W0 stub
MSC	11A51
Status	This copy of the site has no build data. Run `m2 status 0850JJMJYZ`.
Uses	P-adic valuation

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