>>214 補足

こういうときは、英文なんだけど、良い情報がヒットしないな
”Other equivalent definitions use ・・ inverse limits (see §Modular properties)”とあって
Modular propertiesの＜google訳＞は付けたけど
p進のべき級数展開は、昔っからいろんなところで見たけど、なるほど「ニュートン法を使用できます」か
p-adicにすると、微分ができるというのは、雪江 代数3に書いてあったかな
inverse limits (see §Modular properties)
　↓
p進のべき級数展開
を詳しく書いた文献がヒットしない。またあとで

(参考)
https://en.wikipedia.org/wiki/P-adic_number
p-adic number

Definition
There are several equivalent definitions of p-adic numbers. The one that is given here is relatively elementary, since it does not involve any other mathematical concepts than those introduced in the preceding sections. Other equivalent definitions use ・・ inverse limits (see §Modular properties).

Modular properties
The quotient ring {Z}_{p}/p^{n} {Z}_{p}may be identified with the ring {Z}/p^{n} {Z} of the integers modulo p^{n}. This can be shown by remarking that every p-adic integer, represented by its normalized p-adic series, is congruent modulo p^{n} with its partial sum {Σ_{i=0}^{n-1}a_{i}p^{i}, whose value is an integer in the interval [0,p-1]. A straightforward verification shows that this defines a ring isomorphism from {Z}_{p}/p^{n} {Z}_{p} to {Z}/p^{n} {Z}.

The inverse limit of the rings {Z}_{p}/p^{n} {Z}_{p} is defined as the ring formed by the sequences a_{0},a_{1},・・・ such that a_{i}∈{Z}/p^{i} {Z} and {a_{i}≡a_{i+1}{mod {p^{i}}} for every i.

The mapping that maps a normalized p-adic series to the sequence of its partial sums is a ring isomorphism from {Z}_{p} to the inverse limit of the {Z}_{p}/p^{n} {Z}_{p}. This provides another way for defining p-adic integers (up to an isomorphism).

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