Talk:P-adic Expansion Representative of P-adic Number is Unique

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Problem with construct

I'm having difficulty making sense of this:

From P-adic Norm of P-adic Expansion is determined by First Nonzero Coefficient:

$d_l$ is the first index $i \ge m$ such that $d_i \ne 0$
$e_l$ is the first index $i \ge k$ such that $e_i \ne 0$


(a) Definition:Index is a disambiguation page, probably should use Definition:Index Variable of Summation instead. It's always a good idea, when including a link on a page, to check to see where that link actually goes to. The philosophy of this site revolves around the fact that a user can get the precise meaning of anything on any page by following the links.

(b) $d_l$ is not an index of anything, it's an instance of a Definition:Coefficient of Power Series which is indexed by $l$.

(c) Use of Definition:Minimum Element for "first" is inaccurate, as it is not $d_l$ that is the minimum of anything. What it really means is "the first in the sequence such that ..." Recommended not to hunt for a specific link to this -- if we really feel we need to link to a definition of what is meant by "first", then we would need to add it to Definition:Sequence, preferably as a subpage defining "ordinal position in sequence". It's far easier (and probably preferable) to build this concept symbolically, say: $l := \min \set {i: i \ge m \land d_i \ne 0}$


I made similar amendments on P-adic Norm of P-adic Expansion is determined by First Nonzero Coefficient which I am also struggling to get my head round (my failing, I haven't studied this area).


Apart from that, I'm completely lost trying to understand the symbology, probably because I haven't studied the meanings properly. Not quite sure what the equivalence relation is yet that induces the Definition:Equivalence Class under discussion -- perhaps that could be clarified, as can the notation $\mathbf p^n \mathbf a$ which presumably is another instance of such an equivalence class. --prime mover (talk) 04:40, 18 April 2020 (EDT)

Good to get this feedback. I've been bogged down in this area for some time. Neither of the books by Svetlana Katok or Fernando Q. Gouvea provide a fully formal exposition of the connection between $p$-adic expansions and $p$-adic numbers. And in approaching it informally, they do it in different ways.
Both books construct the $p$-adic numbers as the quotient ring of Cauchy sequences by null sequences. The Fernando Q. Gouvea book drops this construction as soon as possible and expresses the $p$-adic numbers as the limit of a $p$-adic expansion in the $p$-adic numbers.
On the other hand, the Svetlana Katok book continues with the construction of the $p$-adic numbers as the quotient ring of Cauchy sequences by null sequences and so the $p$-adic expansions are Cauchy sequences in the rational numbers.
Not surprisingly, these two approaches are equivalent, that is, the same $p$-adic expansion has a $p$-adic number as the limit and is a represenative of the equivalence class that is the $p$-adic number as an element of quotient ring of Cauchy sequences by null sequences.


None of this answers your points, but it helps to clarify in my mind what I'm trying to achieve.


It is straighforward for me to address the index issues above, that was just sloppiness on my part.


The issue on the notation will take more thought. I admit the notation is mine, but I think the notation in the Svetlana Katok book may be more confusing as there is nothing special used. If I were to use the notation of equivalence classes or cosets then the theorem and proof may be quite clumsy. But I probably should go down one of these paths after explaining what the equivalence relation is and what the notation denotes.
--Leigh.Samphier (talk) 23:04, 18 April 2020 (EDT)
Ok, I've now addressesed the index issues above.
Regarding the notation issue, what I need to do is create some theorems that are common to the two approaches used in the books that only require an understanding of $p$-adic expansion and Cauchy sequences without reference to the $p$-adic numbers. For instance, a theorem like:
Let $p$ be a prime number.
Let $\ds \sum_{i \mathop = m}^\infty d_i p^i$ and $\ds \sum_{i \mathop = k}^\infty e_i p^i$ be $p$-adic expansions that differ by a null sequence:
$\ds \sum_{i \mathop = m}^\infty d_i p^i - \ds \sum_{i \mathop = k}^\infty e_i p^i = 0$


Then:
$(1) \quad m = k$
$(2) \quad \forall i \ge m : d_i = e_i$
That is, the $p$-adic expansions $\ds \sum_{i \mathop = m}^\infty d_i p^i$ and $\ds \sum_{i \mathop = k}^\infty e_i p^i$ are identical.
This should be provable with only the rational numbers and the $p$-adic norm on $\Q$ and will require refactoring of some other theorems. --Leigh.Samphier (talk) 02:23, 19 April 2020 (EDT)


That all makes good sense. I will leave off removing the Proofread tag on some of these pages because I am still not quite sure what they mean -- and it will leave it open for other contributors who know their stuff better than me. --prime mover (talk) 04:11, 19 April 2020 (EDT)