Linked Questions

22
votes
6answers
4k views

A resource for learning p-adic numbers

I'm looking for a good resource for learning p-adic numbers. I'm familiar with analysis, topology and overall with noncommutative algebra.
7
votes
2answers
2k views

Inverse limit by example

I'm trying to understand inverse limits. For this I am looking at the example (mentioned in Atiyah-Macdonald, page 102): We start with the topological abelian group $G = \mathbb Z$ (endowed with the ...
9
votes
2answers
900 views

Intuition for limits

My basic intuition for limits/colimits was "limits suck up, colimits suck down". Now, having seen colimits used in presheaf categories, algebraic geometry, and topology, I have much clearer intuition ...
5
votes
2answers
981 views

Idea behind defining a Projective System

What is the idea behind defined a Projective system of Groups/Rings. In our class an example for the Projective system was given by Taking the Ring $\mathbb{Z}/n\mathbb{Z}$ over $\mathbb{N}$. The ...
2
votes
5answers
1k views

What is the remainder when $7^{7^{7^{7…Infinity }}}$ is divided by $5$? [closed]

What is the remainder when $7^{7^{7^{7.........Infinity }}}$ is divided by $5$ ? My try : $7^7$ when divided by $5$ gives the remainder $3$,and similarly, $7^{7^7}$ when divided by $5$ again gives ...
4
votes
3answers
487 views

Colimits glue. What do limits do?

The informal but very useful way to think of colimits is as 'gluing things'. This intuitive view is very helpful to me, and I'd like one for limits aswell, but I haven't stumbled opon such a thing ...
4
votes
2answers
275 views

Understanding the limit in $\mathbb{Z_p}$

Let $A_n = \mathbb{Z}/p^n \mathbb{Z}$ . We define the ring of p-adic integers $\mathbb{Z_p}$ as $$\mathbb{Z_p} := \lim_{\longleftarrow} (A_n , \phi_n)$$ where $\phi_n : A_n \rightarrow A_{n-1}$ is a ...
7
votes
1answer
136 views

What are the ring morphisms $\mathbb{Q}[[X]]\to R$ for a ring $R$?

If $\mathbb{Q}[[X]]$ is the ring of power series over a field $F$, then can we describe ring morphisms $\mathbb{Q}[[X]] \to R$ for rings $R$ in simple terms? I am guessing that a "substitution ...
2
votes
1answer
221 views

What is $\hat{\mathbb{Z}}$?

I have been reading a bit about unramified extensions. If $K$ is $\mathbb{Q}$ or $\mathbb{Q}_p$ (p-adics), then there is a maximal unramified extension $K^{nr}$ of $K$. Then I have read in some notes ...
1
vote
1answer
79 views

$p$-adic Ring extensions vs. “ordinary” Ring extensions

I read about inverse limits in this post, and found the example by Arturo Magidin quite interesting (his "approximate" solution of $x^2 = -1$ in $\mathbb Z$). By his construction we get a Ring which ...