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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Solution of Pell equation over field of p-adic numbers

Right now I am studying Pell equation. Using continued fractions, we can find the solution of Pell equation. Now my question, is it possible for me to find a solution in the field of p-adic numbers ...
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quadratic field extensions of $\mathbb{Q}_p$

Today during class we proved that there were exactly three quadratic field extensions of the $p$-adic number field $\mathbb{Q}_p$. To prove this it was stated that it was enough to look at the group ...
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Which theorem could be used?

I want to write the $p-$adic expansion of $6!$ in $\mathbb{Q}_3$. I have to solve the congruence $x \equiv 6! \pmod {3^n}$, right? I found the following: $$x_0 \equiv 6! \pmod 3 \Rightarrow x_0 ...
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Question about topological properties of $\Bbb{C}_p$

It is known that the structure of $p$-adic integers, $\Bbb{Z}_p$ is homeomorphic to the Cantor set, and $\Bbb{Q}_p$ is homeomorphic to the one-point deleted Cantor set (as I know, I don't certain it.) ...
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no quotient of $\mathcal{O}_{\mathbb{C}_K}$ modulo a proper ideal containing $p$ is a perfect $\mathbb{F}_p$-algebra?

I am reading the Notes on $p$-adic Hodge theory of O. Brinon & B. Conrad . Can someone explains the following things to me? «... no quotient of $\mathcal{O}_{\mathbb{C}_K}$ modulo a proper ideal ...
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Lucas' Theorem for $p$-adic integers?

Does Lucas' Theorem hold for $p$-adic integers? More specifically, does it specifically hold for the case that, given a $p$-adic integer: $$x = x_0 + x_1p + x_2p^2 + \cdots = \sum_{i=0}^\infty ...
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28 views

$\mathbb{Q}_p$ contains a square root of $4p+1$, but not of $4p$.

Let $p$ be a prime, let $\mathbb{Q}_p$ be the field of $p$-adic numbers. I want to show that it contains a square root of $4p+1$, but does not contain a square root of $4p$. I could only manage the ...
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Definition of field of fractions of $p$-adic integers

I am given this definition of the field of fractions of the p-adic integers: $$Q_p=\left\{ \frac{r}{s} \mid r, s \in Z_p, s \neq 0\right\}$$ How can I show that: $Q_p$ consists of the sums of the ...
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The sets are equal

I want to show that $Z_p^*= \{ x \in Z_p | |x|_p=1 \}$. I have tried the following: Let $x \in Z_p^*$, then $x=a_0+a_1p+a_2p^2+ \dots \ \ \ \ \ \ \ \ 1 \leq a_i \leq p-1$. So, $x \in Z_p $. When ...
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Identity of the $p-$norm

A $p-$ norm of $\mathbb{Q}_p$ is a function $||_p: \mathbb{Q}_p \to \mathbb{R}$ $$x \neq 0, x=p^{w_p(x)}u \mapsto p^{-w_p(x)}$$ $$\text{ For } x=0 \Leftrightarrow w_p(x)=\infty \\ p^{-\infty}:=0$$ ...
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Prove identities-p norm

A $p-$ norm of $\mathbb{Q}_p$ is a function $||_p: \mathbb{Q}_p \to \mathbb{R}$ $$x \neq 0, x=p^{w_p(x)}u \mapsto p^{-w_p(x)}$$ $$\text{ For } x=0 \Leftrightarrow w_p(x)=\infty \\ p^{-\infty}:=0$$ ...
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Power series in $\mathbb{Q}_5$

Could you help me to find the first five positions of the power series in $\mathbb{Q}_5$ of $\frac{1}{2}$? How can I do this? Is there a general formula?
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28 views

What does the inequality stand?

The additive $p-$ adic valuation of $\mathbb{Q}_p$: $$w_p: \left\{\begin{matrix} \mathbb{Q}_p \rightarrow \mathbb{Z} \cup \{\infty\}\\ p^m u \mapsto m\\ 0 \mapsto \infty \end{matrix}\right.$$ ...
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55 views

How can we show that it is an integer 5-adic number?

Show that the number $\frac{3}{8}$ is an integer $5$-adic and calculate the first five positions of its power series in $\mathbb{Q}_5$. Could you explain me how we can conclude that $\frac{3}{8}$ is ...
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Ring of the integer $p$-adic numbers $\mathbb{Z}_p$

Let the ring of the integer $p$-adic numbers $\mathbb{Z}_p$. Could you explain me the following sentences? It is a principal ideal domain. $$$$ The function $\epsilon_p: \mathbb{Z} \to ...
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Completed group ring

Let $p$ be a prime number. Consider $\mathbb{Z}_p$ the ring of $p$-adic integers. If $G$ is a profinite group, define $\mathbb{Z}_p[[G]]$ to be the inverse limit $\lim\mathbb{Z}_p [G/U]$ where $u$ ...
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Tate's thesis - continuous map from a local field to circle group

I am currently reading Decomposition of Unitary Representations defined by a discrete subgroups of nilpotent groups, by C.C. Moore. It is metioned that if $\mathbb{K}$ is a $p$-adic field in his ...
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38 views

Does a generalization of the Teichmuller-character for non-prime arguments exist?

Rereading an older article on Fermat-quotients in which I'd applied some p-adic-rationale I find now, that my method for the representation of bases $b$ which allow high fermat-quotients ...
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25 views

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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33 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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36 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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46 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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73 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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19 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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58 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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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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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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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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46 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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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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35 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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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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50 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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43 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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31 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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57 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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63 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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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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115 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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52 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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33 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 ...