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.

learn more… | top users | synonyms

-1
votes
1answer
53 views

P-adic coefficient series [closed]

Let $f$ be: $$f(x)=\sum_{n=0}^{\infty} a_nx^n$$ Such that $a_n \in \mathbb{Z_p}$ and $a_n \to 0$ We denote $|f|=\max|a_n| \space $and $\deg(f)=\max\{n:|a_n|=|f|\}$.$$$$ ($a$) Prove that if ...
1
vote
0answers
39 views

Torsion points on elliptic curves, $E^1$

Let $E$ be an elliptic curve over $\mathbb{Q}_p$ and let $E(\mathbb{Q}_p) \supset E^0(\mathbb{Q}_p) \supset E^1(\mathbb{Q}_p) \supset \cdots$ be its $p$-adic filtration, where $E^n(\mathbb{Q}_p) = \{P ...
3
votes
1answer
34 views

Examples where there is no power integral basis

Let $A$ be a completion discrete valuation ring with quotient field $K$, $L/K$ finite and separable, and $B$ the integral closure of $A$ in $L$. Let $P, \mathfrak P$ be the unique maximal ideals of ...
1
vote
1answer
25 views

Rescaling of Ternary quadratic forms

I was reading about the Hilbert residue symbol, and the discussion of it starts out with the assumption that we can reformat any ternary quadratic form over the integers into the form ...
3
votes
1answer
47 views

About p-adic numbers

I'm studying the dual group of the diadic rationals, $\widehat{\mathbb{Z}[1/2]}$, where the dual is the dual of Pontryagin of $\mathbb{Z}[1/2]$. In some papers says that $\widehat{\mathbb{Z}[1/2]}$, ...
2
votes
1answer
49 views

$p$-adic numbers and projective coordinates

Let $E/\mathbb{Q}_p$ be an elliptic curve and let $E^0(\mathbb{Q}_p)$ denote its nonsingular points. We accept that $E^0(\mathbb{Q}_p)$ is a subgroup of $E(\mathbb{Q}_p)$. Then let ...
2
votes
1answer
53 views

elliptic curves over $\mathbb{Q}_p$

Let $E: Y^2 = X^3 + AX + B$ be an elliptic curve over $\mathbb{Q}_p$, i.e. $A,B \in \mathbb{Q}_p$ and $4A^3 + 27B^2 \neq 0$. Then, according to page 47 of Cassels' Lectures on Elliptic Curves, if ...
0
votes
0answers
40 views

Cassel's book on Elliptic Curves

Let $E/\mathbb{Q}_p$ be an elliptic curve. Then for $n \geq 1$, let $E_n(\mathbb{Q}) = \left\{P \in E(\mathbb{Q}_p) : \dfrac{x(P)}{y(P)} \in p^n \mathbb{Z}_p\right\}$. According to Cassels in Lectures ...
1
vote
1answer
36 views

Does the p-adic rationals have isolated points?

Does $\mathbb Q_p$ have isolated points? I think that it doesn't,but i cannot prove it. Any help?Thank you!
3
votes
1answer
89 views

History of $p$-adic numbers

I'm interested in learning about the historical motivation and development of $p$-adic numbers. I haven't been able to find any books on the topic. I'd appreciate any references, including to more ...
3
votes
1answer
40 views

Can one check by hand whether the Tate module of an elliptic curve is semi-simple

Let $E$ be an elliptic curve over $\mathbb Q$, and $\ell$ a prime number. Then, the $\ell$-adic Tate module $V_\ell(E)$ of $E$ is semi-simple as a $\mathbb Q_\ell$-representation of ...
3
votes
0answers
21 views

properties of certain semigroup action on $\mathbb{Z}/p\mathbb{Z}$

Suppose we have a polynomial $f \in \mathbb{Z}/p\mathbb{Z}[x]$, $f(x) = x^2 - x$. We are interested in elements $n \in \mathbb{Z}/p\mathbb{Z}$ such that after repeated application of f they ...
3
votes
1answer
71 views

Why exponential function on p-adic numbers is meaningless?

In the notes, page 3, it is said that $e^{2\pi i r y}$ is meaningless if $y$ is a general p-adic number. Why exponential function on p-adic numbers is meaningless? Thank you very much.
0
votes
1answer
65 views

Questions about p-adic expressions.

In the notes, page 3, it is said that in $2$-adic and $3$-adic expansion, we have $$ \frac{21}{50} = \frac{1}{2} + 2 + 2^2 + \cdots \tag 1 $$ $$ \frac{21}{50} = 2\cdot 3 + 3^2 + 3^6 + \cdots \tag 2 ...
5
votes
0answers
28 views

A p-adic integral

Let $(K,||)$ be a finite extension of $\mathbb{Q}_p$ of degree $d$ such that the restriction of $||$ to $\mathbb{Q}_p$ is the usual p-adic absolute value. Endow $GL_n(K)$ with the unique Haar measure ...
2
votes
1answer
29 views

$p$-adic expansion

I have just touched on this topic, please guide me along. If I have a prime number $p=10^{10}+19$, and a $p$-adic number $\alpha=\frac{16}{17}$. How do I derive its $p$-adic expansion? Thanks in ...
0
votes
0answers
34 views

$P-adic$ number theory problem

Let $F(x,y,z)=5x^2+3y^2+6z^2+8xy+6xz$. Find all the rational integers $x,y,z$ such that they are not all divisible by $7$ and that $F(x,y,z)=0 mod(7^2)$. Hint: Use Hensel's lemma. Need help. ...
2
votes
0answers
25 views

$\mathbb{Z}_p[\text{PGL}_2(\mathbb{F}_{p^n})]$-modules

Let $p$ be prime. Let $\mathbb{Z}_p$ be the $p$-adic integers. I'm intersted in $\mathbb{Z}_p[\text{PGL}_2(\mathbb{F}_{p^n})]$-modules of low rank over $\mathbb{Z}_p$ (rank $2$ is already very ...
1
vote
1answer
28 views

Continuous p-adic function similar to q-adic norm

Given two distinct primes $p,q$, I am looking for some notion of a $q$-adic valuation of $p$-adic numbers. Obviously, I can define $f:\Bbb{Q}_p \to \Bbb{Q}_p$ by $f(x) = \begin{cases}|x|_q & x\in ...
0
votes
1answer
29 views

Cauchy sequence in $\mathbb{Q}_p$ implies its p-absolute value is cauchy in $R$

Actually, I don't understand why $\{ a_{n}\} \in \mathbb{Q}_{p}$ is cauchy implies $|a_{n}|_{p} \in \mathbb{R}$ is cauchy. Could anyone give me a hint for understand this?
1
vote
1answer
23 views

What does it mean by “formal identity” in the sixth line from the top on page 79 from Koblitz's p-adic Numbers, p-adic Analysis, and Zeta-Functions.

What does it mean by "formal identity" in the sixth line from the top? The text comes from page 79 from Koblitz's p-adic Numbers, p-adic Analysis, and Zeta-Functions. Any explanation is appreciated. ...
0
votes
0answers
22 views

When is it easy to write down the Bhargava S-factorial?

In Manjul Bhargava's The Factorial Function and Generalizations he motivates a new type of factorial $n!_S$ using by generalizing three theorems: For $k, l \in \mathbb{Z}$, we have $k! \times l!$ ...
0
votes
0answers
33 views

Galois representation associated to a number field

I think I'm missing something completely trivial. I want to know how to compute the Galois representation associated to an extension of $p$-adic fields. Let $p$ and $q$ be odd prime numbers. Fix ...
1
vote
0answers
55 views

Frobenius action on $\overline{\mathbb Q_p}$

Let $p$ be a prime number and let $F_p$ be the Frobenius automorphism of $\overline{\mathbb F_p}$. Given an explicit element $x $ of $\overline{\mathbb Q_p}$, how do I compute $F_p(x)$? Does it even ...
0
votes
0answers
28 views

p-adic analysis problem

Let $x \in \mathbb{Q}$. Show that if $\alpha \in \mathbb{Z}$ is an integer such that $|\alpha - x|_{p} \leq p^{-i} $ for some $i\in \mathbb{N}$, then there exists $\alpha' \in \{ 0,1,2, \ldots, ...
1
vote
2answers
59 views
2
votes
1answer
68 views

Proof of Hasse's principle for quadratic equations

I am currently tackling the following problem. Problem Consider the equation $x^2 = q, $ where $ q \in \mathbb{Q}$. Show this has a rational solution $x$ in $\mathbb{Q}$ if and only if there are ...
1
vote
2answers
45 views

Proving the p-adic numbers $\mathbb{Q}_p$ form a field

I am trying to prove that $\mathbb{Q}_p$ forms a field. However, I am unsure of the best way to go about proving it. If I work with the power series representation of p-adic numbers I run in to ...
0
votes
0answers
24 views

Proving the set $\mathbb{Z}_2$ of 2-adic integers is compact. [duplicate]

I am currently working on the problem: Prove the set $\mathbb{Z}_2$ of 2-adic integers is compact My idea is to prove this via sequential compactness. So far, I have considered a sequence ...
3
votes
2answers
71 views

Direct proof of compactness of $\mathbb{Z}_p$

Let $\mathbb{Z}_{p}$ be completion of $\mathbb{Z}$ with respect to $p-$norms. Actually I know that $\mathbb{Z}_{p}$ is bijective to Cantor set, which is compact, therefore by homeomorphism, it is also ...
0
votes
0answers
21 views

Representing numbers in unique p-adic to matrix representation, is there a way?

What I am wanting to do, if it is possible is find unique matrix representations for the p-adic representation of numbers. So for example, say I have the number 1365 = 3*5*7*13. Now I could take ...
0
votes
1answer
56 views

7-adic series expansion of square root of 2

Given the sequence $\{ a_n\}$ defined by the (positive and $a_n < 7^n$) solutions of the congruence $x^2 \equiv 2 \mod 7^n$ and $a_{n+1}\equiv a_n \mod 7^n$. e.g. the first one is $a_1 =3$ the ...
1
vote
3answers
58 views

p-adic numbers, fractions in $\mathbb{Z}_p$

How can we write $1/3$ in $\mathbb{Z}_5$ as series? We can write it as $\frac{1}{5-2}=-\frac{1}{2-5}=-(2+2*5+2*5^2+\dots)$ but $-2$ are negative coefficients...Please explain if there is a flaw in my ...
1
vote
1answer
54 views

Is $f(x)=x^{4}-2x^2 +3$ Eiseinstein in 2-adic $\mathbb{Q}_{2}$?

I think it is because $|1|_{2}=1$, $|2|_{2}=2^{-1}\leq 1$ and $|3|$,where 3 is prime I am using the following Eisenstein criterion for $f(x)=a_{n}x^{n}+...+a_{0}$: $|a_{n}|=1$, $|a_{0}|=prime$ and ...
0
votes
1answer
23 views

Possible Quadratic extensions of $\mathbb{Q}_{2}$ are 1-1 with $\mathbb{Q}_{2}^{*}/(\mathbb{Q}_{2}^{*})^{2}$

Why do they have to be units? $\mathbb{Q}_{2}[\sqrt{2}]$ is a quadratic extension but $|2|_{2}=\frac{1}{2}\neq 1$. Where do I need them to be units below? ...
1
vote
2answers
55 views

A quadratic equation over $\mathbb{Q}_p$

Suppose we have the equation $x^2+x+1$ over the field $\mathbb{Q}_p$. is it possible to determine for what primes $p$ the equation has solutions? I tried to see whether this is related to what $p$ is ...
0
votes
1answer
40 views

relation between $p$-adic numbers and congruence relations modulo p

Set $p$ prime. Take $f_1, f_2 \in \mathbb{Z}$ with $f_1 f_2$ square free and $p \nmid f_1, f_2$ and $\alpha \in \mathbb{Q}_p$ such that $|\alpha|_p = 1$ (i.e. $\alpha$ is a $p$-adic unit). Show that ...
1
vote
1answer
37 views

What is the p-adic valuation of elements of $\mathbb{Q}_p$ not in $\mathbb{Q}$.

In Cassels book "Lectures on Elliptic Curves", he defines the $p$-adic integers as: $\quad \mathbb{Z}_p = \{\alpha \in \mathbb{Q}_p \mid |\alpha|_p \leq 1\}$ He latter states that the $p$-adic ...
6
votes
1answer
103 views

Problems that are made easy by using p-adic numbers

Does anybody know of elementary problems that can be be solved using the p-adics? Solutions are preferred.
1
vote
0answers
45 views

Open Ball under the p-adic Norm

I'm trying to figure how, if it's even possible, to draw an open ball using the p-adic norm. My definition of the p-adic norm I'm using is: $ \lvert x \rvert_p $ = $p^{-ord_px}$ if $x \neq 0$ and ...
3
votes
2answers
47 views

$2$-adic inverse of $3$

Performing long division, I calculated the $2$-adic inverse of $3$ to be $$1-2+4-8+...$$ Then I noticed that I could get the same result in a more sleek way by noting that $$\frac 1 3 = \frac ...
1
vote
1answer
25 views

Ramification and roots of unity in complete discrete valuation rings.

Let $\mathcal{O}$ be a complete discrete valuation ring with algebraically closed residue field $k$ of characteristic $p>0$. Let $\pi\in \mathcal{O}$ generate the maximal ideal and suppose ...
5
votes
2answers
85 views

Degree of closure of $\mathbb{Q}_p$

In order to prove that algebraic closure of $\mathbb{Q}_p$ is infinite, I took the polynomial $x^n-p$ with $n>1$ over $\mathbb{Q}_p$ to show that this eqaution has no solution for infinite cases to ...
6
votes
1answer
103 views

What does root of unity in $\mathbb{Z}_p$ look like?

Let $p$ be an odd prime. Then by Hensel's lemma it's clear that $\mathbb{Z}_p $ contains all $p-1$th root of unity which reduces to $1$, $2$, ... , $p-1$ in $\mathbb{F}_p$. My question is do we know ...
0
votes
1answer
75 views

What is $e, \pi, \ln 2,…$ etc in p adic?

What is $e, \pi, \ln 2,...$ etc in p adic? And how to flip digits of decimal points? Does p-adic have their own constants? 10 adic base.
5
votes
2answers
102 views

Open problems involving p-adic numbers

I am in my final year of my undergraduate degree, and I'm doing a project on p-adic numbers, and in particular, trying to find Galois groups of simple extensions of $\mathbb{Q}_p$ (this is a Galois ...
2
votes
1answer
46 views

$p$-part of cyclotomic character

Let $K$ be a number field, $\bar K$ a separable closure and $p$ a rational prime and assume that $p \not= char(K)$. Consider the extension $$K(\mu_{p^\infty}) \mid K$$ which is obtained by adjoining ...
3
votes
1answer
57 views

Number of field extensions of $\mathbb{Q}_p$

If I know the index $(\mathbb{Q}_p^{\times} : (\mathbb{Q}_p^{\times})^n)$ for some $n \in \mathbb{N}$, is it possible to know how many field extensions of $\mathbb{Q}_p$ of degree $n$ there are? This ...
0
votes
1answer
64 views

Embedding into $p$-adic complex numbers

As I'm reading notes about the Leopoldt conjecture, the following question came to my mind: Let $\mathbb{C}_p$ be the $p$-adic complex numbers, i.e. the completion of the algebraic closure of the ...
4
votes
1answer
50 views

The unramified quadratic extension of $\mathbb{Q}_2$

I know that there are $7$ field extensions of $\mathbb{Q}_2$ of degree $2$ (this follows from Hensel's lemma) and I think these are $$\mathbb{Q}_2(\sqrt{2}), \mathbb{Q}_2(\sqrt{3}), ...