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.

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Prove some properties of the $p$-adic norm

I need to prove that the p-adic norm is an absolut value in the rational numbers, by an absolut value in a field I mean a function that goes from $K \to \mathbb{R}_{\ge 0}$ such that: I)$|x|=0 ...
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31 views

p-adic numbers and GCD

Given two numbers $a,b \in \mathbb{Z}$, how do we prove that the $p$-adic number of $\gcd(a,b)$ is the same as the minimum for the $p$-adic number of $a$ and the $p$-adic number of $b$? Does this ...
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33 views

Why $\mathbb{Q}_p^{ur} \neq \widehat{\mathbb{Q}_p^{ur}}$?

Why is $\mathbb{Q}_p^{ur}$ not complete? And is there a criterion to know when $K^{ur} = \widehat{K^{ur}}$ ? (where $K$ is a p-adic field, i.e. a field of characteristic 0 that is complete with ...
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P-adic Lie groups - Representation theory

I am quite familiar with the Representation Theory for locally compact groups and nilpotent Lie groups. I want to start with the study of $p$-adic Lie groups representation theory, in particular ...
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33 views

Showing that Q is not complete with respect to the p-adic absolute value

I am looking at some notes that give an example of a Cauchy sequence that doesn't converge in $\mathbb{Q}$ with respect to the $p$-adic absolute value. Their example is to let $1 < a< p-1$ and ...
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62 views

Show that i is an element of the p-adic integers if and only if p congruent to 1 mod 4

This exercise was given in a graduate course on Local Class Field Theory. We want to prove that $i\in \mathbb{Z}_p$ (the $p-$adic integers) if and only if $p\equiv 1 \mod 4$. For $\Rightarrow$, we ...
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18 views

Non-standard extensions of $p$-adic fields

Does there exist a non-standard extension of a non-Archimedean field (such as the construction $*\mathbb{R}$ out of $\mathbb{R}$ or the surreals $\mathbb{S}_\mathbb{R}$, not to mention their ...
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55 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 ...
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15 views

Strong Approximation for adelic quaternionic groups

Let $H$ be a definite quaternion algebra defined over $\mathbb{Q}$, unramified at $p$, and ramified at $\infty$. Denote by $D$ the multiplicative group $H^\times$ divided by its center. In this ...
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37 views

Surjectivity of p-adic representation

Let $E/\mathbb{Q}$ be an elliptic curve. For a prime $p$, we have the mod $p$ representation \begin{equation*} \bar{\rho}_{E,p}: G_{\bar{\mathbb{Q}}/\mathbb{Q}} \rightarrow Aut(E[p]) \end{equation*} ...
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42 views

Which p-adic groups are simply-connected?

Suppose that we are working over a nonarchimedean local field $F$, for instance $\mathbb{Q}_p$. Which semisimple algebraic groups (or Lie groups) over $F$ are simply-connected? In particular, I am ...
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44 views

The space $C_c^\infty(\mathbb{Q}_p^*)$ of smooth compactly supported functions on $\mathbb{Q}_p^*$

Let $p$ be prime. Let $\mathbb{Q}_p^*$ be the multiplicative group of the field of $p$-adic numbers. We call a function $f:\mathbb{Q}_p^*\rightarrow\mathbb{C}$ smooth if it is invariant under ...
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51 views

p-adic cubic root

Let $p$ be prime such that $p\equiv 2\bmod 3$. Show that for every $a\in \mathbb Z,p\nmid a$ there is a $x\in \mathbb Z_p$, where $\mathbb Z_p$ is the field of the p-adic integers, such that $x^3=a$. ...
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Torsion of finitely generated $\mathbb Z_ \ell$-module finite?

Let $M$ be a finitely generated $\mathbb Z_ \ell$-module, where $\ell$ is a prime number and $\mathbb Z_\ell$ is the ring of $\ell$-adic integers. Let $T$ be its torsion submodule. Is $T$ finite? In ...
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28 views

Surjective $p$-adic representation implies trivial $p$-primary part.

Let $E/\mathbb{Q}$ be an elliptic curve. We know that by Serre in the non-CM case, for $p\geq5$, $$\rho_p:Gal(\bar{\mathbb{Q}}/\mathbb{Q})\rightarrow Aut(T_p(E))$$ is surjective iff $$ ...
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34 views

How do I interpret the order that an $p$-adic $L$-series vanishes to?

I know how to find the the order of vanishing for a complex $L$-series $L(E,1)$. I'm looking at an example: ...
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69 views

finding “exp(1)” in the p-adic numbers

Can anyone give me some kind of description when, in $\mathbb{C}_p$(the completion of the algebraic closure of the p-adic numbers), there is an element $x$ which satisfies $ Log_p(x) = 1, $ and in ...
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45 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 ...
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41 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 ...
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28 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 ...
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55 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 $\widehat{\mathbb{Z}[1/2]}$, ...
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61 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 ...
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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 ...
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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 ...
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38 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!
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109 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 ...
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45 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 ...
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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 ...
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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.
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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 ...
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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 ...
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1answer
31 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 ...
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36 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. ...
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$\mathbb{Z}_p[\text{PGL}_2(\mathbb{F}_{p^n})]$-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 ...
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30 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 ...
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30 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?
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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. ...
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35 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 $k! \times l!$ ...
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35 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 ...
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58 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 ...
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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, ...
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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 ...
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52 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 ...
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25 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 ...
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76 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 ...
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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 ...
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61 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 ...
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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 ...
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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 ...