Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [p-adic-number-theory]

In mathematics the $p$-adic number system for any prime number $p$ extends the ordinary arithmetic of the rational numbers in a way different from the extension of the rational number system to the real and complex number systems.

1
vote
0answers
21 views

Problem with proof of Strong Triangle Inequality of P-adic numbers

So I have seen many proofs of the Strong Triangle Identity for the p-adic numbers. Namely, that: $$ \| x + y \|_p \leq \text{max}\left( \|x\|_p, \|y\|_p\right) $$ All such proofs use the assumption ...
0
votes
1answer
35 views

Algebraic closure of p-adic rationals, $\overline{\mathbb Q}_p$, and its completion, $\Omega_p$, are not locally compact

Trying to show $\overline{\mathbb Q}_p$ and $\Omega_p$ are not locally compact. I can prove it by showing that the unit sphere is not locally compact. That is to say, any sequence on the unit sphere ...
0
votes
0answers
75 views

Find $x$ such that $3x+2^{\nu_2(x)}\in\left\{\frac{-(2^n+3)}{2^m9}:n\in\Bbb N_{>0},m\in\Bbb Z\right\}$ [on hold]

Find the set $X$ of all $x$ such that $3x+2^{\nu_2(x)}\in 2^mY$ Where $Y=\left\{\dfrac{-(2^n+3)}{9}:n\in\Bbb N_{>0}\right\}$ $2^{\nu_2(x)}$ is the highest power of $2$ that divides $x$. $Y$ set ...
2
votes
0answers
67 views

limits of powers of a matrix with entries in p-adic numbers; galois representation of finite fields

Let $u$ be a $d \times d$ matrix with coefficients in $\mathbb{Q}_\ell$, the $\ell$-adic numbers. Let $n \in \widehat{\mathbb{Z}}$. Something I'm reading claims (and says it's easy and leaves the ...
1
vote
1answer
42 views

A question about formal power series and Amice transform

Let's call $L$ the continuous dual space of $C(\mathbb{Z_p},\mathbb{Q_p})=\{f:\mathbb{Z_p}\rightarrow\mathbb{Q_p}\mid f\text{ continuous}\}$, so $L$ consists on the continuous linear functions from $C(...
0
votes
2answers
43 views

Question in p-adic integration (Igusa type)

I am trying to learn how to solve Igusa type local zeta function. Ex. $$\int_{\mathbb{Z}_{p}}||x^3,x^2y,y^2||d\mu(x,y)$$ A nice method I was recently introduced to was to substitute $x=a+px'$ and $y=...
0
votes
1answer
27 views

Is a singleton in a p-adic space connected?

I want to use a theorem with a p-adic space: The image by a continuous epimorphism of a connected space, is itself connected. Correct me if I'm wrong, but I think the fact that every p-adic field is ...
3
votes
0answers
38 views

Why does the image of $ord_p$ form an additive subgroup of $(1/n)\mathbb Z$?

Let $K$ be a field extension of the p-adic rationals $\mathbb Q_p$. The image of $K^\times$ under the valuation map $ord_p(x)=-\frac 1 n\log_p|\mathbb N_{K/\mathbb Q_p}(x)|_p$ is contained in $(1/n)\...
1
vote
1answer
131 views

What is the sequence of accumulation points in the 2-adic space, of the Collatz graph?

In the orbit of the function $3x+2^{\nu_2(x)}$ through "accumulation points" of the Collatz graph I have: $?\mapsto\dfrac{-\langle2\rangle\cdot\{5,7\}}{9}\mapsto\dfrac{-\langle2\rangle}{3}\mapsto \...
1
vote
1answer
23 views

Injection of $\mathbb{Z}$ into a p-adically complete abelian group

If $M$ is a p-adically complete abelian group, so that it's a $\mathbb{Z}_p$-module, and we have an injective homomorphism $\phi: \mathbb{Z} \hookrightarrow M$, is it true that the induced ...
0
votes
1answer
29 views

Is $\overline{\langle2\rangle\cdot(4\Bbb Z+1)}=\langle2\rangle\cdot(4\Bbb Z_2+1)$ in $\Bbb Z_2$?

Is $\overline{\langle2\rangle\cdot(4\Bbb Z+1)}=\langle2\rangle\cdot(4\Bbb Z_2+1)$ in $\Bbb Z_2$? $\langle2\rangle$ is the set of powers of $2$ and $\cdot$ is the straightforward dot product. I ...
1
vote
0answers
12 views

Double coset decomposition of $N_n \backslash SL_2(\mathbb Q_p) / N_n$

By Bruhat decomposition, $B \backslash SL_2(\mathbb Q_p) / B$ consist of two elements where $B$ is the group of upper triangular matrices. And from this we get the structure of double cosets $N \...
1
vote
1answer
55 views

All numbers close enough to 1 are squares

Behind this provocative title is a precise question. I have been working on some books and papers claiming the following. Let $F$ be a number field, and denote by $p$ its finite places. Here is the ...
0
votes
2answers
53 views

Notation in $p$-adic integers

When we write $\mathbb{Z}_3$, does it mean $\mathbb{Z}/3\mathbb{Z}$? Also, does $3\mathbb{Z}_3$ mean $0 \pmod 3$?
2
votes
1answer
38 views

tamely totally ramified extensions and the equation $x^e-pu=0$

Considering $\mathbb Q_p$ the p-adic rationals; Tamely totally ramified extensions are obtained by adjoining solutions of the equation $x^e-pu=0$, where $e$ is the index of ramification and $u\in \...
0
votes
1answer
50 views

Any element in $K/\mathbb Q_p$ can be generated by some $\pi \in K$

following a sentence from "p-adic Numbers, p-adic Analysis and Zeta-functions" by Neal Koblitz, page 66: Let $\pi \in K$ where $K$ is an extension field of $\mathbb Q_p$ (the p-adic rationals) and $...
0
votes
2answers
47 views
4
votes
1answer
180 views

The multiplicative group of units in $\mathbb{Z}_{p}$ is isomorphic to $\mathbb{Z}_{p} \times C_{n}$ with $n=\max\{p-1,2\}$.

Problem. (a) Write down explicitly the rules for addition and multiplication in $\mathbb{Z}_{p}$. (b) Show that $(\mathbb{Z}_{p^{i}},\varphi_{ij})$ where $\varphi:\mathbb{Z}_{p^{j}} \to \mathbb{...
4
votes
2answers
64 views

degree extension over filed of $p$-adic numbers

Let $K = \mathbb{Q}(\theta)$ be a numberfield and $[K:\mathbb{Q}]=n$. When $\mathbb{Q}_p$ is the field of $p$-adic numbers and $K_p=\mathbb{Q}_p(\theta)$, what about $[K_p : \mathbb{Q}_p]$?
3
votes
1answer
117 views

Galois group of unramified extension of $\mathbb{Q}_p$ is cyclic

I have seen that there for every $f$ there exists a unique unramified extension to $\mathbb{Q}_p$ of degree $f$, and it is $\mathbb{Q}_p(\delta)$ where $\delta$ is a primitive $(p^f-1)$-root of unity. ...
3
votes
2answers
106 views

solving a $p$-adic integration involving maximum function

I am always struggling when it comes to dealing with maximum functions. I am trying to find the solution to this integral $$\int_{\mathbb{Z}_p^3}||xy,xz,yz||_p^sd\mu(x,y,z),$$ where $||xy,xz,yz||_p^...
3
votes
2answers
89 views

Totally ramified extension of $\mathbb{Q}_{p}$ which is not of a form $\mathbb{Q}_{p}(\sqrt[n]{pu})$

It is known that a finite extension $K/\mathbb{Q}_{p}$ is totally ramified if and only if $K = \mathbb{Q}_{p}(\alpha)$ where $\alpha$ is a root of Eisenstein polynomial. Is there any totally ramified ...
1
vote
1answer
19 views

Haar Measures and Embeddings of $\nu$-adic integers

Let $\nu\geq4$ be any composite integer, and let $d\in\left\{ 2,\ldots,\nu-1\right\}$ be any non-trivial divisor of $\nu$. Since $d\mid\nu$, note that any sequence $\left\{ \mathfrak{y}_{n}\right\} _{...
1
vote
3answers
104 views

problem in $p$-adic integration

I am working on the $p$-adic integration and I am trying to find how to integrate $$\int_{\mathbb{Z}_p^2}||x,y||_p^sd\mu (x,y),$$ where $d\mu$ is the haar measure and $||x,y||_p^s=\sup\{|x|_p^s,|y|_p^...
4
votes
1answer
202 views

Completing Algebraic Integers into Squares

Let $L/K$ be an extension of number fields with Galois closure $E$, and let $\theta \in \mathcal{O}_L \setminus \{0\}$. Let $\Sigma_E$ be the set of primes of $E$, let $S' \subset \Sigma_E$ be a ...
0
votes
1answer
29 views

Affine Change-of-Variables Formula in $p$-adic Haar measure integration.

Let $p$ be a prime number, let $V$ be an arbitrary non-empty subset of $\mathbb{Z}_{p}$, and let $d\mu$ be the Haar measure on $\mathbb{Z}_{p}$ subject to the normalization $\int_{\mathbb{Z}_{p}}d\mu=...
0
votes
1answer
92 views

The isomorphism of $\mathbb{Z}_{p}[x] /\left(x^{n}-1\right)$ using Hensel's Lemma

I an trying to prove the following.Let p:prime, $n\in\mathbb{N}$ with $(n,p)=1$ ,and $x^{n}-1=f_{1}(x) \cdot \ldots \cdot f_{r}(x) \quad\left(f_{1}(x), \dots, f_{r}(x) \in \mathbb{Z}_{p}[x]\right.$ ...
0
votes
2answers
95 views

How would one explain the concept of a p adic number in layman's terms?

The concept of a p adic number is something that I haven't quite been able to grasp. Heretofore, I can comprehend the fact that the p adic number system essentially extends the conventional arithmetic ...
1
vote
0answers
53 views

Field Extension $ \mathbb{Q} \subset \mathbb{Q}_p $ infinite

Let $\mathbb{Q}_p$ be the $p$-adic rational field. I want to verify that the field extension $\mathbb{Q}_p/ \mathbb{Q}$ is infinite therefore $\dim_{\mathbb{Q}}(\mathbb{Q}_p) = \infty$. My ...
7
votes
1answer
123 views

Topological Algebraic Independence of power series

Let $p$ be a prime number, let $x$ be a variable, and consider two power series over the ring $\mathbb{Z}_p$ of $p$-adic integers: $a(x):=\underset{n\geq 1}{\sum}{\frac{p^n}{n!}x^n}=px+\frac{p^2}{2}x^...
-1
votes
1answer
51 views

Non-zero ideals in ${\mathbb{Q}}_p$ are $p^n{\mathbb{Q}}_p$, $n\in\mathbb N_0$

How do I show that every non-zero ideal in ${\mathbb{Q}}_p$ is of the form $p^n{\mathbb{Q}}_p$ for some $n \in \mathbb{N}_0$, and investigate if ${\mathbb{Q}}_p$ is a principal ideal domain? If it ...
2
votes
0answers
48 views

Bound for roots of a polynomial with coefficients in a non-Archimedean valued field

Is there any bound for the valuation of the roots of a given polynomial with coefficients in an algebraically closed non-Archimedean valued field? Any reference or insight would be appreciated.
2
votes
0answers
40 views

Algebraic closure of $\mathbb{Q}$ in $\mathbb{Q}_{p}$

Let $(K, |\cdot |)$ be a (discrete) valuation field, and $(\widehat{K}, |\cdot|)$ be its completion. Then we can think about the algebraic (or separable) closure of $K$ in $\widehat{K}$, which is ...
0
votes
1answer
69 views

Does $ \ (\pi \mathbb{Z}[\zeta_p])^2 \subset S \ \ $ hold ?, p-adic numbers

$\underline{\text{p-adic Numbers}}:$ Consider the cyclotomic extension $k=\mathbb{Q}_p(\zeta_p)$ of the p-adic field $\mathbb{Q}_p$ and let $ \mathbb{Z}_p[\zeta_p]$ be the ring of integer of $\mathbb{...
0
votes
0answers
34 views

Let $p$ be a prime and $a$ be a primitve $p^N\text{th}$ root of 1 in $\mathbb{Q}_p$. find $|a-1|_p$ [duplicate]

This is Q7 from p.73 of Koblitz, p-adic numbers, p-adic analysis and zeta functions which is available here http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.461.4588&rep=rep1&type=pdf ...
0
votes
1answer
37 views

convergent sequence in set of p-adic numbers

How can we set up a sequence in $\mathbb{Q}$ which is convergent in $\mathbb{Q_5}$, but not convergent in $\mathbb{Q_7}$?
2
votes
1answer
76 views

Algebraic closure of $\mathbb{Q}$ in $\mathbb{Q}_p$

Algebraic elements in $\mathbb{Q}_p$ are well studied and there was a question on mathoverflow about information regarding this. I want to know following information: Let $K_p=\overline{\mathbb{Q}}...
1
vote
0answers
66 views

Change of variables in $p$-adic integration

I'm trying to understand Serre's mass formula, but it's not very clear to me what does it mean to integrate over a subset of $L$, or $K^n$, and in particular why it holds an analogue of the usual ...
2
votes
0answers
36 views

thinking a $p$-adic integer in different ways

In Serre's course in arithmetic, $p$-adic integers are defined as follows: let $A_n:=\mathbb{Z}/p^n\mathbb{Z}$. There are natural (ring) homomorphisms $\phi_n:A_n\rightarrow A_{n-1}$. The ring of $p$-...
0
votes
0answers
20 views

Structure of Hecke algebra $\mathcal{H} = C_{c}^{\infty}(\mathrm{GL}(2, F))$.

Let $F$ be a non-archimedean local field. If $K = \mathrm{GL}(2, \mathcal{O}_{F})$ is a maximal compact subgroup of $G = \mathrm{GL}(2, F)$, then we understand the structure of $\mathcal{H}_{K} = C_{c}...
1
vote
3answers
91 views

limit in p-adic number system

Please give me a hint for limit of $lim_{n\to\infty}3^{2^n}$ in $\mathbb Q_5$. First, new absolute value $|\cdot|'$ on $\mathbb Q$ is defined as the following: For $\frac{n}{m}\in \mathbb Q$,$$|\...
0
votes
0answers
43 views

Proof of $\mathbb{Q}_{p} \not\simeq \mathbb{Q}_{q}$ for $p\neq q$ - proof verification

I want to know whether my proof of $\mathbb{Q}_{p}\not\simeq \mathbb{Q}_{q}$ for $p\neq q$ primes is right or not. (I know that this question is already asked several times on MSE) We can assume $q\...
0
votes
1answer
35 views

This is about p-adic norm. When $\left|\left(\frac{u}{v} \right)^N\pm 1 \right|_p<1$?

This is about p-adic norm. If $x=[\left(\frac{u}{v} \right)^N\pm 1], \ u,v \in \mathbb{Z}, \ N \in \mathbb{N}$, then which conditions on $u,v, N$ ensures that $\left|\left(\frac{u}{v} \right)^N\pm ...
0
votes
0answers
15 views

Which condition on $x=[\left(\frac{u}{v} \right)^N\pm 1], \ N \in \mathbb{N}$ ensures that $\left|\left(\frac{u}{v} \right)^N\pm 1 \right|_p<1$?

Consider the p-adic norm $|.|_p$ defined by $|x|_p=p^{-n}$, where $x=\frac{a}{b}p^n, \ (a,b)=1$ for some non-zero rational number $x$. Which condition on $x=[\left(\frac{u}{v} \right)^N\pm 1], \ N \...
3
votes
0answers
49 views

Classification of (finite dimensional) admissible representations of $F^{\times} = \mathrm{GL}(1, F)$.

Let $F$ be a non-archimedean local field. In Bump's textbook, there are two kinds of 2-dimensional such representations: $$ t\mapsto \begin{pmatrix} \xi(t) & \\ & \xi'(t)\end{pmatrix} $$ for ...
2
votes
1answer
37 views

Formal power series rings and p-adic solenoids

For any prime $p$, the ring of $p$-adic integers can be generated as the quotient of the formal power series ring $\Bbb Z[[x]]/(x-p)$. My questions: If we instead use $\Bbb Q[[x]]/(x-p)$, do we get ...
4
votes
0answers
105 views

Relationship between $p$-adic numbers and analytic continuation of $1+x+x^2+x^3+…$

The infinite sums $1 + 2 + 4 + 8 + ...$ $1 + 3 + 9 + 27 + ...$ $1 + 5 + 25 + 125 + ...$ $1 + 7 + 49 + 343 + ...$ ... of powers of primes do not converge in the usual sense. However, by analytically ...
1
vote
0answers
26 views

equality in $p$-adic analysis

I'm reading a paper of recurrences sequences and I could not understand why the equality mentioned in the underlined lines in the image. I think that it has to do with $p$-adic analysis. If someone ...
2
votes
2answers
65 views

normal subgroup basis of $\mathrm{GL}(n, \mathbb{Q}_{p})$

Consider the group $G=\mathrm{GL}(n, \mathbb{Q}_p)$. This group is locally compact and totally disconnected, and we have a basis of open subgroups given by $$ K(p^m) = \left\{ \begin{pmatrix} a&b\...
0
votes
0answers
32 views

deformations of $p$-divisible groups

I've been wanting to learn more about the deformation theory of $p$-divisible groups and was looking for some references. I have looked into stuff like the Serre-Tate theorem with Katz's Serre-Tate ...