>>84 補足

Z＾（hat付き）と、Z＾(1)（hat付き）と
どちらも、巡回群の逆系を作って、それを利用して逆極限を作る
群論的にも圏論的にも、両者は関連している
だからこそ、 「Z＾(1) (円分物)」という記号を使っているのだろう

さて、Z＾関連で、前スレ https://rio2016.5ch.net/test/read.cgi/math/1623019011/930 より MITの講義
https://math.mit.edu/classes/18.782/lectures.html
LECTURES MIT Arithmetic Geometry
https://math.mit.edu/classes/18.782/LectureNotes4.pdf
Introduction to Arithmetic Geometry Fall 2013
Lecture #4 Andrew V. Sutherland

Example 4.7. We have the following p-adic expansion in Z_7:
2 = (2, 2, 2, 2, 2, . . .)
2002 = (0, 42, 287, 2002, 2002, . . .)
-2 = (5, 47, 341, 2399, 16805, . . .)
2^-1 = (4, 25, 172, 1201, 8404, . . .)
√2 = ((3, 10, 108, 2166, 4567 . . .)
　　=(4, 39, 235, 235, 12240 . . .)
2^(1/5) = (4, 46, 95, 1124, 15530, . . .)
You can easily recreate these examples (and many more) in Sage. To create the ring of 7-adic integers, just type Zp(7).
By default Sage will use 20 digits of p-adic precision, but you can change this to n digits using Zp(p,n).

https://math.mit.edu/classes/18.782/LectureNotes7.pdf Lecture #7 Introduction to Arithmetic Geometry Fall 2013
Remark 7.19. Everything we have done here applies more generally to commutative rings.
For example, Zp is the completion of Z with respect to the p-adic absolute value | |p on Z,
as we will see in the next lecture.
（ #8 Hensel's lemma ）
(引用終り)

つづく