Definition talk:P-adic Valuation/Integers

Further thoughts on this.

Since, in the "also known as", it is stated that this is "usually" defined on the rationals, it make sense to make that one the main page. Then the merge with Definition:Multiplicity of Prime Factor, which can be defined as the restriction of this to the integers, can be implemented as a second definition on that page, rather than this one. --prime mover (talk) 02:01, 25 August 2017 (EDT)

Merge with Definition:Prime Decomposition/Multiplicity

Regarding the suggestion of a merge Definition:Prime Decomposition/Multiplicity it should be noted that the multiplicity of $p$ is only defined if $p$ is a prime factor of an integer, where as the $p$-adic valuation is always defined for every integer. The multiplicity of $p$ is never 0, but the $p$-adic valuation can be. This subtle difference means that the two definitions are not the same. --Leigh.Samphier (talk) 12:52, 7 April 2021 (UTC)

Okay, no merge -- but what we may want to do is to add some "warning" page which specifically highlights the difference between them, and perhaps "also see" somne implications of that.

I confess I get lost in the tiny details. Might be worth adding some examples pages to illustrate instances of these various objects and constructs. --prime mover (talk) 13:52, 7 April 2021 (UTC)

You could create a theorem that shows that for any integer the $p$-adic valuation equals the multiplicity if $p$ divides the integer and 0 otherwise. This states explicitly the relationship between the two definitions. The page could be used to emphasise that the multiplicity for $p$ is not defined if $p$ does not divide an integer. --Leigh.Samphier (talk) 10:09, 8 April 2021 (UTC)