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

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 $\displaystyle \sum_{i \mathop = m}^\infty d_i p^i$ and $\displaystyle \sum_{i \mathop = k}^\infty e_i p^i$ be $p$-adic expansions that differ by a null sequence:
 * $\displaystyle \sum_{i \mathop = m}^\infty d_i p^i - \displaystyle \sum_{i \mathop = k}^\infty e_i p^i = 0$


 * Then:
 * $(1) \quad m = k$
 * $(2) \quad \forall i \ge m : d_i = e_i$


 * That is, the $p$-adic expansions $\displaystyle \sum_{i \mathop = m}^\infty d_i p^i$ and $\displaystyle \sum_{i \mathop = k}^\infty e_i p^i$ 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)