5
votes
2answers
103 views

$\mathbb{Q}_p\otimes_{\mathbb{Q}} \mathbb{Q}_q$ and $\mathbb{Z}_p\otimes_{\mathbb{Z}} \mathbb{Z}_q$

Let $p, q$ be prime numbers which may or may not be distinct. Let $\mathbb{Q}_p$ be the field of $p$-adic numbers. Let $\mathbb{Z}_p$ be the ring of $p$-adic integers. We define similarly ...
2
votes
2answers
114 views

$p$-adic valuation.

Let $\alpha_1,\alpha_2\in \mathbb Z_p$ such that $v_p(\alpha_1)<v_p(\alpha_2).$ How to prouve that $v_p(\alpha_2-\alpha_1)=v_p(\alpha_1)$ ? I think this is a stupid question but I'm really ...
2
votes
1answer
45 views

How to calculate the derivative of a function in $\mathbb{Q}_p$?

On Wikipedia it is stated that the function $$ f:\mathbb{Q}_p\to \mathbb{Q}_p $$ with $f(x)=(1/|x|_p)^2$ if $x\neq 0$ and $f(0)=0$ is differentiable and its derivative is the zero-function. How ...
4
votes
1answer
241 views

Multidimensional Hensel lifting

I have a question about a practical application of (some) generalised form of Hensel's Lemma. I cannot find it stated in an appropriate form in Bourbaki or anywhere else, so here goes ... Let $p$ be ...
1
vote
1answer
101 views

Valuation rings of complete non-archimedean fields which are not local

I would like to know how are the valuation rings of complete non-archimedean fields which are not local. Take, for example, $\mathbb{C}_p$, and its valuation ring, ...
5
votes
2answers
388 views

Question about $p$-adic numbers and $p$-adic integers

I've been trying to understand what $p$-adic numbers and $p$-adic integers are today. Can you tell me if I have it right? Thanks. Let $p$ be a prime. Then we define the ring of $p$-adic integers to ...
9
votes
1answer
125 views

diagonalizing a matrix over the $\ell$-adics

Let $M$ be a $2 \times 2$ matrix with coefficients in $\mathbb{Z}_{\ell}$ whose characteristical polynomial is $$ P(T) = T^2- (a+d) T + (ad-bc). $$ I've encountered the following assertion: If ...
11
votes
2answers
301 views

Why is $O_K\otimes \mathbb{Z}_p\cong \oplus_{\mathfrak{p}|p}O_{K,\mathfrak{p}}$?

In my old number theory notebook this is stated as a fact. However, I ran into problems when I tried to prove it. First let me state the (supposed) theorem accurately: Theorem (?) Let $K$ be a ...
6
votes
1answer
296 views

Integral closure of p-adic integers in maximal unramified extension

Let $\mathbb Q_p$ be the field of p-adic numbers, and let $\mathbb Q_p^{\text{unr}}$ be maximal unramified extension in some algebraic closure of $\mathbb Q_p$. My understanding is that $\mathbb ...
4
votes
1answer
116 views

Subgroups of index 3 in $1+p\mathbb{Z}_p$

Let $p$ be a prime. I'm trying to compute the subgroups of index $3$ in $\mathbb{Q}_p^\times$ to enumerate some cyclic extensions using CFT. I've essentially reduced the problem down to finding the ...