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This article's motivation

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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)[reply]

Definition of p-adic numbers

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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 (talkcontribs) 10:44, 30 November 2020 (UTC)[reply]

@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 such that 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 such that 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)[reply]

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 (talkcontribs) 10:37, 1 December 2020 (UTC)[reply]

@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)[reply]

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

@:@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)[reply]

@Joe in Australia: "the -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 -adic valuation. This is not the case, so the second "-adic" should be kept. The phrase "is a measure of the size of " doesn't make sense without first specifying what "size" refers to; in fact the size is defined using the -adic valuation, so we can't talk about size before saying what the -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, (and every other power of ) divides the rational number since there exists a rational number such that . Eric Rowland (talk) 23:39, 3 December 2020 (UTC)[reply]

@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)[reply]

Article name

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

Unless there is additional discussion, I'll go ahead and rename the article. Eric Rowland (talk) 17:17, 19 June 2022 (UTC)[reply]
This is now done. Eric Rowland (talk) 18:17, 21 June 2022 (UTC)[reply]

Sentence about

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@Nomen4Omen: How is the sentence "Thereby, as in the projectively extended real line, and for ." related to this article? There are no real numbers in the article (other than rational numbers), and the preceding formula never involves an addition of the form since and are rational numbers. Eric Rowland (talk) 20:29, 1 April 2022 (UTC)[reply]

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)[reply]