Talk:P-adic valuation

This article's motivation

In creating this stub, I recognize that its mathematical content is already much better covered by sections of the articles Valuation (mathematics) and p-adic number, it's given a place in Multiplicity, and it's very similar to 2-order. I admit that it's undesirable to duplicate coverage yet again.

However, the p-order is such an elementary concept, and so commonly used, that I think it will prove useful to have an article specifically on the concept, rather than treating it is a mere by-product of a more powerful formalism, or a mere stepping stone towards more widely used machinery. (Of course, this article should still expand on applications and generalizations.)

So, I'll be creating links to this stub/article in a few other articles, and I hope other editors will tolerate the duplication issue as being, to some extent, a necessary evil in mathematics. Ultimately, when this article is filled out, it should be able to help out other articles per Wikipedia:Summary style. Melchoir (talk) 03:03, 2 March 2008 (UTC)

Definition of p-adic numbers

It is incorrect to say that p-adic valuations are only defined for non-zero integers. The article itself clearly and correctly says otherwise. Please don't revert this correction. — Preceding unsigned comment added by Joe in Australia (talk • contribs) 10:44, 30 November 2020 (UTC)

@Joe in Australia: How is it possible that you come to the conclusion that the article had said:

"that p-adic valuations are only defined for non-zero integers".

What it said was:

"the p-adic valuation of a non-zero integer n is the highest exponent $\nu$ such that $p^{\nu }$ divides n."

This is 100 % correct — and additionally much more precise than yours.

Where do you get this only from? Your fantasy? I don't see it!
(As you know there is a continuation which extends the definition, so it is two sentences. I guess there should be no need to repeat this.) Do you have the problem that you are unable to perceive 2 sentences? I donnow.

And how can you get the impression that p-adic numbers would be defined (see your subtitle #Definition of p-adic numbers) by your statement:

"... the p-adic order or p-adic valuation of a number n is the highest exponent $\nu$ such that $p^{\nu }$ divides n."

Your numbers are completely undefined. What do you mean by number? Integer, rational, real, complex, p-adic (a circular definition??) ? And what does "divide" mean in your edit? –Nomen4Omen (talk) 17:53, 30 November 2020 (UTC)

IMO the implication of the original phrasing is that the p-adic valuation is only defined for non-zero integers. I appreciate that the first paragraph is meant to be a quick definition rather than a full one, but it seems confusing. How about this:

In number theory the order or valuation of a p-adic number, n, commonly denoted as ν_p(n, is a measure of the size of n. For non-zero integers, ν is the highest exponent such that p^ν divides n. If n/d is a rational number in lowest terms, so that n and d are coprime, then ν_p(n/d) is equal to ν_p(n), if p divides n, or −ν_p(d) if p divides d, or to 0 if it divides neither. The p-adic valuation of 0 is defined to be infinity.

That does leave out the valuation of irrational p-adic numbers, which ought to be addressed in the body of the article, but I feel that it's more complete and less confusing. As for the ultimately circular nature of the definition, I think it comes with the territory: p-adic values are only defined for p-adic numbers, which are those that can have p-adic values. — Preceding unsigned comment added by Joe in Australia (talk • contribs) 10:37, 1 December 2020 (UTC)

@Joe in Australia: Your new proposal somehow looks really better. But still I have a big problem:

The article's title is „P-adic order“, AND NOT „P-adic number“, as you appear to insist. There is indeed available an article named „P-adic number“! And as far as I know WP, this latter article would have to somehow define p-adic numbers, whereas the current article „P-adic order“ would have to define some p-adic order or maybe p-adic valuation. Possibly —at least in the lead— for the integers and rationals, and —at least in the lead— totally irrespective of the p-adic numbers.

I would propose that you take this under consideration. –Nomen4Omen (talk) 17:25, 1 December 2020 (UTC)

A lot of people know what integers are. Far fewer people know what $p$-adic numbers are. In the interest of broad accessibility, it is much better to start with the definition for integers. Extensions to $p$-adic numbers (and algebraic extensions, etc.) can be discussed later in the article. Eric Rowland (talk) 18:25, 1 December 2020 (UTC)

@:@Eric Rowland: Would it be better if I moved the word "p-adic" so it said "the p-adic order or valuation of a number, n, commonly denoted as ν_p(n), is a measure of the size of n"? Using the word "number" without qualification is at least sloppy, as Nomen4Omen said, but as the explanation would immediately go on to explain the p-adic order of integers and rationals it doesn't necessarily leave people in the position of thinking that, e.g., the p-adic order is a quality of real numbers. Joe in Australia (talk) 07:42, 2 December 2020 (UTC)

@Joe in Australia: "the $p$-adic order or valuation of a number" makes it look like "valuation of a number" can be used on its own to refer to the $p$-adic valuation. This is not the case, so the second "$p$-adic" should be kept. The phrase "is a measure of the size of $n$" doesn't make sense without first specifying what "size" refers to; in fact the size is defined using the $p$-adic valuation, so we can't talk about size before saying what the $p$-adic valuation is. The third issue is "number" vs. "integer". I don't see any reason to use "number" when "integer" is more precise. For rational numbers, the "highest exponent" definition is actually incorrect, because of the definition of "divides"; for example, $2^{9}$ (and every other power of $2$) divides the rational number $5/3$ since there exists a rational number $x$ such that $2^{9}x=5/3$. Eric Rowland (talk) 23:39, 3 December 2020 (UTC)

@Joe in Australia: The topic of the article is p-adic order which is a topic of Euclid's time and not p-adic number which is a topic of the 19th century. Issue resolved in this sense. –Nomen4Omen (talk) 13:05, 12 December 2020 (UTC)

Article name

It seems this article should be renamed "$p$-adic valuation" from "$p$-adic order", since the term "$p$-adic valuation" is 2 to 5 times more common than "$p$-adic order". On English Wikipedia, 14 articles contain the string "adic order" while 40 articles contain "adic valuation". Google search produces 5590 results for "$p$-adic order" and 30900 results for "$p$-adic valuation". Eric Rowland (talk) 16:54, 1 April 2022 (UTC)

Unless there is additional discussion, I'll go ahead and rename the article. Eric Rowland (talk) 17:17, 19 June 2022 (UTC)

This is now done. Eric Rowland (talk) 18:17, 21 June 2022 (UTC)

Sentence about $x+\infty$

@Nomen4Omen: How is the sentence "Thereby, as in the projectively extended real line, $x+\infty =\infty$ and $\infty >x$ for $x\in \mathbb {R}$." related to this article? There are no real numbers in the article (other than rational numbers), and the preceding formula $\nu _{p}(r+s)=\min {\bigl \{}\nu _{p}(r),\nu _{p}(s){\bigr \}}$ never involves an addition of the form $x+\infty$ since $r$ and $s$ are rational numbers. Eric Rowland (talk) 20:29, 1 April 2022 (UTC)

Sorry, I didn't see the already present footnote «<ref name="infty">with the usual [[Extended real number line#Arithmetic operations|rules for arithmetic operations]]</ref>».

Done.

Go ahead for better formulations.

–Nomen4Omen (talk) 21:19, 1 April 2022 (UTC)