Definition talk:P-adic Number

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Definition of P-adic Numbers

This page is meant to reflect the two definitions of the P-adic Numbers from the books:

Katok defines the field of $p$-adic numbers to be the construction of the quotient of the Cauchy Sequences by the Null Sequences of the rational numbers with respect to the $p$-adic norm. After a while the book drops this construction once the necessary properties of the $p$-adic numbers for the rest of the book are stated.

Gouvêa on the other hand, shows that the completion exists (the same construction as in Katok) and then refers to any completion of the rational numbers with respect to the $p$-adic norm as the $p$-adic numbers.

These two definitions are not equivalent, although Katok's definition is a specific example of a completion covered by Gouvêa's definition. And every completion under Gouvêa's definition is isometrically isomorphic to Katok's definition.

This is where I got bogged down on $p$-adic numbers. But with some distance, I've a little more clarity.

In my opinion, the correct definition is that of Gouvêa and so this page should simply state that.

So I suggest the following changes:

  • Rename Definition:P-adic Number/P-adic Norm Completion of Rational Numbers $\to$ Definition:P-adic Number.
  • Rename Definition:P-adic Number/Quotient of Cauchy Sequences in P-adic Norm $\to$ Definition:P-adic Numbers as Quotient of Cauchy Sequences.
  • Fix up all links accordingly.
  • Add an Also see link on Definition:P-adic Number $\to$ Definition:P-adic Numbers as Quotient of Cauchy Sequences, and vice-versa.

Not 100% convinced of the name Definition:P-adic Numbers as Quotient of Cauchy Sequences. So open to any other suggestions.

Then I can address specific comments raised on the other pages:

Let me know if anyone has any other thoughts.

--Leigh.Samphier (talk) 02:46, 10 January 2021 (UTC)

Definition:P-adic Numbers as Quotient of Cauchy Sequences should either be a proof page P-adic Numbers as Quotient of Cauchy Sequences (as in: "a p_adic number can be constructed as a quotient of cauchy sequences like this" sort of thing, or we have a "Definition 1" and "Definition 2" sort of thing. The second approach (as you suggest) is questionable, as the two forms are not equivalent.
Similarly for the other things. We are in danger of over-complexifying the definition structure by defining ever more intricate subpage names. If a "P-adic number" is a well-defined entity, we can define it using the technique "Definition 1", "Definition 2" etc., and then prove the things are the same using an equivalence proof.
I also have difficulty with a definition page which first starts statements justifying the validity of the construction. The reader gets bogged down in a lot of preliminary material which needs to be fought through before getting to the actual definition. What we have been doing throughout the rest of the site is to place the definition as the first thing the user encounters, and placing the justification for the constructions into pages invoked in "Also see", usually just by including the link itself (appropriately named), but occasionally with a few words, ",... which proves that the construction is justified", or whatever can be included to make it clear.
I understand the temptation to pile as much information into a definition page as possible, but we really want to keep a definition page as clean and neat as possible. We have not always managed to achieve this, but it's our aim. Hence I deeply mistrust sections called "Notes" or "Comments" and so forth, because it can lead to information and results that ought to be on separate pages.
As for Definition:P-adic Metric/P-adic Numbers and Definition:P-adic Norm/P-adic Numbers, we are nearly where we want to be, but I am unhappy with Definition:P-adic Norm. We have our 2 equivalent definitions for the concept on the rational numbers, and then a completely different definition for it on the p-adic numbers, both at the same level, making it confusing. The answer should not be "oh, let's just make the definition 1 and definition 2 pages level 4 headings" because (shudder). Perhaps we should make the rational numbers definition Definition:P-adic Norm/Rational Numbers and then make the top-level page a parent page which would then not be linked to directlt, but only contain the transcluded instantiations: one for rationals, one for p-adics (and leaving ourselves open for further implementations of p-adic norms on other structures still). This would require a bunch of relinking every where Definition:P-adic Norm appears to Definition:P-adic Norm on Rational Numbers which redirects to Definition:P-adic Norm/Rational Numbers. Same would apply to Definition:P-adic Metric. Laborious, but it's nothing we haven't done before. --prime mover (talk) 09:09, 10 January 2021 (UTC)

Proposal for Minor Refactor of P-adic Number

Currently the definition for the Definition:P-adic Norm on P-adic Numbers is indirect and requires you to go to the page Definition:P-adic Number to get the detail of the definition of the Definition:P-adic Norm on P-adic Numbers.

This was highlighted when I looked at adding the Definition:P-adic Norm on P-adic Numbers to the overall Definition:Norm page. The definition was not at all enlightening.

So I looked at breaking the definition for the $p$-adic numbers into 2 parts. The first part being the definition of the field (see User:Leigh.Samphier/Definition:Field of P-adic Numbers) and the second part being the definition of the norm on the $p$-adic numbers (see User:Leigh.Samphier/Definition:P-adic Norm on P-adic Numbers)

The new definition for the $p$-adic numbers defines the $p$-adic numbers as the valued field constructed from the field of $p$-adic numbers and the $p$-adic norm on the $p$-adic numbers (see User:Leigh.Samphier/Definition:P-adic Numbers)

I have also pulled out the definition of a $p$-adic number in to its own page as an element of the field of $p$-adic numbers (see User:Leigh.Samphier/Definition:P-adic Number)

So I am proposing that the existing pages

Definition:P-adic Number
Definition:P-adic Numbers
Definition:P-adic Norm/P-adic Numbers

be refactored into the pages:

User:Leigh.Samphier/Definition:Field of P-adic Numbers
User:Leigh.Samphier/Definition:P-adic Norm on P-adic Numbers
User:Leigh.Samphier/Definition:P-adic Number
User:Leigh.Samphier/Definition:P-adic Numbers

I have already gone through the existing links to Definition:P-adic Number and Definition:P-adic Numbers and updated links so that all references to the valued field of $p$-adic numbers are linked to Definition:P-adic Numbers and all references to a $p$-adic number are linked to Definition:P-adic Number.

So I am looking for feedback and comment on the proposal. --Leigh.Samphier (talk) 19:26, 23 May 2022 (UTC)

My real concern is over having two closely related, but different, pages Definition:P-adic Number and Definition:P-adic Numbers differing by only a single letter in their page names.
Maybe it would be better to rename Definition:P-adic Numbers as Definition:Valued Field of P-adic Numbers --Leigh.Samphier (talk) 07:24, 24 May 2022 (UTC)
We have the same page for "natural number" and "natural numbers", and we use that page when discussing the "Set of natural numbers", But when we start discussing the algebraic properties of numbers, we go to things like: Definition:Ring of Integers, for example. Yours is a good plan: keep "P-adic Number but more P-adic Numbers to Field of P-adic Numbers when you need specifically to discuss its properties as a field. --prime mover (talk) 17:45, 24 May 2022 (UTC)