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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Isogeny of elliptic curves over $p$-adic field

If $K$ is a $p$-adic field, and $E_q$ and $E_{q'}$ are the corresponding Tate curves for $|q|,|q'|<1$, why does $E_q$ and $E_{q'}$ being isogenous imply that there are integers $A$ and $B$ such ...
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82 views

P-adic expansion of rational number

Maybe this is a silly question but I really can not see how to get a p-adic expansion of a rational number. I do know the case of for an integer but how can I extend to the rational number case. If we ...
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$p$-adic valuation of harmonic numbers

For an integer $m$ let $\nu_p(m)$ be its $p$-valuation i.e. the greatest non-negative integer such that $p^{\nu_p(m)}$ divides $m$. Let now $H_n=1+\dfrac{1}{2}+ \cdots+ \dfrac{1}{n}$. If $H_n=\...
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$K|\mathbb{Q}_p$ un-ramified if and only if $(d_K)=(1)$: help with a passage

I need some help in th last passage of this proof: Suppose $K|\mathbb{Q}_p$ is un-ramified and of degree $n$. then $K=\mathbb{Q}_p(\alpha)$, where $\alpha$ can be taken to be an integral unit in $\...
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Definition of singular solution of $f(\mathbf{x}) = 0$ in $p$-adic integers

Let $f(\mathbf{x}) \in \mathbb{Z}[x_1, ..., x_n]$ and consider the equation $f(\mathbf{x}) = 0$. I am wondering what exactly does it mean by "the equation $f(\mathbf{x}) = 0$ has a non-singular ...
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Un-ramified extension of $\mathbb{Q}_p$. A clarification on the construction

I'm following the proof given in Koblitz's book which roughly speaking builds the un-ramified extension of degree $f$ of $\mathbb{Q}_p$ as $\mathbb{Q}_p(\alpha)$, where $\alpha$ is a root of the lift ...
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How to write the Adeles over $\mathbb{Q}(i)$?

What do these adeles look like over $\mathbb{Q}(i)$? These are simply fractions where the numerator and denominator are allowed to have the number $i = \sqrt{-1}$. The elements look like: $$ \left\...
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113 views

Finding maximum number of factors in n!

I am quite new to $v_p()$ problems, and would like to know if anyone prove that $v_n(n!)\le n/2$? Basically, what I mean is that prove that for all positive integers $n$, the amount of factors of $n$ ...
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If $B$ is an abelian group, then is $B{\otimes}_{\mathbb Z}{\mathbb Z}_p$ isomorphic to ${\varprojlim}B/p^{n}B$?

I could get the easy map from $B{\otimes}_{\mathbb Z}{\mathbb Z}_p$ to ${\varprojlim}B/p^{n}B$ but I could not find the map in the opposite direction. Please help me. Thank you!!
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Finding an example of a non-rational p-adic number

We know that every rational number can be written as a $p$-adic integer with expansion $\sum\limits_{n=-m}^\infty a_n p^n$, where $a_n\in\{0,\dots,p-1\}$ and $m\in\mathbb{N}$; therefore there exists ...
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Showing the positivity of $p$-adic density of zeroes of a polynomial

Let $f \in \mathbb{Z}[x_1, \ldots, x_n]$ and $p$ be a prime. Let $\nu_t(p)$ denote the number of solutions $\mathbf{x} \in ((\mathbb{Z}/p^t \mathbb{Z}))^*)^n$ to the congruence $$ f( \mathbf{x} ) \...
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anisotropic Forms over 2-adic integers

I would like to know, if there is a 4 dimensional anisotropic quadratic form over the 2-adic Integers $\mathbb{Z}_2$, that satisfies the following property: It is in diagonal form and 2 does not ...
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How to make sense of the binomial coefficient over $p$-adic integers?

I recently asked this question: Show that $y^2=x^3+1$ has infinitely many solutions over $\mathbb Z_p$. and now I'm trying to make sense of the first answer that was posted. It said that I should show ...
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51 views

$p$-adic logarithm is injective if $p > 2$?

Define the $p$-adic logarithm$$\log_p(1 + x) = \sum_{i = 1}^\infty (-1)^{i-1}x^i/i.$$I know that $\log_p$ is a homomorphism from $U_1$ to the additive group of $\mathbb{Q}_p$, where $U_1$ is the ...
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Show that $y^2=x^3+1$ has infinitely many solutions over $\mathbb Z_p$.

I first compared it with how I would solve this over the real numbers. You would say: $y^2=\alpha$ has a solution for all $\alpha>0$, of which there are infinitely many. $x^3+1>0$ for all $x&...
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72 views

$p$-adic logarithm, $|\log_p(1 + x)|_p = |x|_p$?

Define the $p$-adic logarithm$$\log_p(1 + x) = \sum_{i =1}^\infty (-1)^{i-1}x^i/i.$$How do I see that if $p > 2$ and $|x|_p < 1$, then $|\log_p(1 + x)|_p = |x|_p$?
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Algebraic Closure and $p$-adic completion: do they commute? [duplicate]

I know that the algebraic closure of $\mathbb Q_p$, which I'll denote $\overline{\mathbb Q_p}$, is not metrically complete: there are $p$-adic Cauchy sequences that do not converge. (The example I ...
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29 views

Definition for non-degenerate module

[QUESTION] If $R$ be a ring, what is the meaning of a non-degenerate $R$-module? In a previous question post at (What is a non-degenerate module?), some experts said that if $M$ is a $R$-module such ...
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1/p in p-adic number system?

to expand $1/p$, I tried first letting $1/p = a+b*p+c*p^2+d*p^3+...$ and it is $1=a*p+b*p^2+...$ but I guess there's no way to make the equality hold. it's somewhat similar to dividing by 0. is it ...
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Property of valuation in $\Bbb{Z}_p$

Let $v$ denote the $p$-adic valuation in $\Bbb{Z}_p$. Let $a_1,a_2,a_3$ be three elements of $\Bbb{Z}_p$. Then I want to show that $$\min \{ v(a_1), v(a_2), v(a_1+a_2+a_3)\}=\min \{v(a_1),v(a_2),v(...
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valuation of a particular element in $\Bbb{Z}_p$

Consider $x \in \Bbb{Z}_p$. Then I want to find the valuation of $(1+p)^x-1$. I think that $val_p((1+p)^x-1)=1+val_p(x)$. Is this right? Actually I want to prove that $min\{val_p(1+p)^x-1, val_p(...
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About 2-adic representation of integers

How would I express -3 in 2-adic representation? Is it just revercimal calculation of binary expression of -3? like: -3 = -11 in binary, so using revercimal, -11. in binary = 01. ?
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38 views

Compute Limit of p-adic Cauchy Sequence

This has really been irking at me and I really should be able to do this but for some reason I can't so I'll ask on here. It is easy to compute the rational "equivalent" of a Cauchy sequence of the ...
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1answer
111 views

Projective limit involving p-adic numbers

Let $p$ and $q$ be distinct primes. What is the projective limit $$\varprojlim \mathbb R^2 / (p^n \mathbb Z \times q^n \mathbb Z)?$$ That's an exercise from Robert's book A Course in p-adic ...
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1answer
31 views

Do the adeles $\mathbb{A}$ have the same cardinality as the real numbers $\mathbb{R}$?

My question today is about the Adeles and whether they have the same cardinality as the real numbers. Certainly $\mathbb{R}$ has more elements than $\mathbb{Z}$. This is by Cantor diagonalization. ...
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43 views

$p$-adic sine series

This is a quick question about the domain of convergence of $p$-adic sine series. We define the $p$-adic sine function by the following power series $\sin_p(X) = \sum\limits_{n=0}^\infty (-1)^n\frac{...
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Prove that the canonical p-adic expansion of $a\in\mathbb{Q}_p$ terminates

I am having trouble getting started with the following problem: Prove that the canonical p-adic expansion of $a\in\mathbb{Q}_p$ terminates (so $a_i = 0$ for all $i \geq N$) if and only if a is a ...
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58 views

7-adic expansion of a rational number

I know that every rational number has a unique 7-adic expansion, now I need help proving that $7/36=\sum\limits_{i=0}^nn7^n$ as a 7-adic integer. I tried using properties of this fraction, like adding ...
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1answer
35 views

How to calculate $\log_p x$ in p-adic analysis?

I'm studying p-adic analysis and recently I've learned about the p-adic logarithm function but I can't understand very well how the process of calculating the value should be done. As an exercise I'm ...
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49 views

Fast way to check if finite extension is unramified?

Consider the $5$-adic quadratic extension $\mathbb{Q}_{5}(\sqrt{5})/\mathbb{Q}_{5}$. I want to check if this extension is unramified, where unramified means that the corresponding extension of ...
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37 views

Solution of a equation in $\Bbb{Z}_p$

Let $m \in \Bbb{Z}_p$ be fixed. Let $a_1,...a_l$ be fixed integers. I am trying to find out solutions of the equation $m=x_1^{a_1}...x_l^{a_l}$ where $x_1,...,x_l\in \Bbb{Z}_p$. Here $x_1,...x_l$ are ...
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Quadratic extensions of p-adic rationals

In Alain Robert's A Course in p-adic Analysis, the author uses Hensel's Lemma to analyze quadratic extensions of $\mathbb{Q}_p$. He wants to calculate the index of $(\mathbb{Q}^*_p)^2$ in $\mathbb{Q}^*...
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Prove that the sequence $(p^n)^{∞}_{n=1}$ for p a prime is a null sequence with respect to $| · |_p$.

Prove that the sequence $(p^n)^{∞}_{n=1}$ for p a prime is a null sequence with respect to $| · |_p$. Here is what I have: a null sequence is a sequence that maps to zero. $| · |_p=p^{-vp(\cdot)}$ ...
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A theorem on p-adic power series

This is a question about a proof of a theorem in many books on $p$-adic numbers and I don't seem to understand one of the directions. The theorem is Let $f(X) = \sum\limits_{n=0}^\infty a_nX^n \in \...
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Relation between Bombieri theorem and p-adic squares

Koblitz states in his book on p-adic numbers on page 84: Suppose that $\alpha \in \mathbb Q$ is such that $1 + \alpha$ is the square of a nonzero rational number $a/b$. Let $S$ be the set of all ...
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Tautological line bundle over rational projective space

Is the tautological line bundle over $\mathbb{Q}P^{n}$ a non trivial bundle? Here, $\mathbb{Q}P^{n}$ has the natural topology induced from the standard topology of $\mathbb{Q}$ as a subset of $\...
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Square of a p-adic number is an integer.

This is a question for an assignment, so just some pointers in the right direction would be great. Suppose that $n \in \mathbb Z$ and that $\alpha \in \mathbb Q_p$ satisfies $\alpha^2 = n$. Prove ...
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Understanding proof that $\mathbb{Q}_{p}(\zeta)/\mathbb{Q}_{p}$ is unramified for $(n,p)=1$.

Problem Consider the extension $\mathbb{Q}_{p}(\zeta)/\mathbb{Q}_{p}$, where $\zeta$ is a $n$-th primitive root of unity and $(n,p)=1$. I want to show that $\mathbb{Q}_{p}(\zeta)/\mathbb{Q}_{p}$ is ...
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n-th power residues in p-adic fields

How to see that number of n-th power residue classes of a p-adic field K (elements of $K^*/K^{*n}$) is finite? I know how to prove using Hensel lemma that all p-adic integers sufficiently close to 1 ...
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Prove that $\mathbb{Z}_p = {a\in\mathbb{Q}_p:|a|_p\leq 1}$.

Let $\mathbb{Z}_p= {a\in\mathbb{Q}_p : a=\sum_{i=0}^{\infty}d_ip^{i},0\leq d_i\leq p-1\;\forall\;i\geq 1}$ be a subset of $\mathbb{Q}_p.$ In other words, $\mathbb{Z}_p$ is the set of all p-adic ...
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Minimal polynomial has coefficients in the valuation ring?

Question Let $K$ be a complete nonarchimedean valued field with valuation ring $\mathcal{O}$ and let $L/K$ be a finite extension. Let $\alpha$ be an element of $L$ and $f\in K[x]$ its minimal ...
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Cuspidal and Supercuspidal representation

Let $G$ be an algebraic group over a field $F$, and let $(\pi,V)$ be a smooth $G$-representation over an algebraic closed field $k$. Then $\pi$ is called a CUSPIDAL representation if $r(V)=0$ for any ...
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Let $p_1,p_2\in\mathbb{Z}$ be distinct prime numbers. Show that $|\cdot|_{p_1}\not\sim|\cdot|_{p_2}$

Let $p_1,p_2\in\mathbb{Z}$ be distinct prime numbers. Show that $|\cdot|_{p_1}\not\sim|\cdot|_{p_2}$ by finding a sequence in $\mathbb{Q}$ that is Cauchy with respect to $|\cdot|_{p_1}$ but not Cauchy ...
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Prove that the sequence $(s_n)$ in $\mathbb{Q}$,

Prove that the sequence $(s_n)$ in $\mathbb{Q}$, where $$\sum_{i=-m}^{n}d_i,p^i$$ with $0\leq d_i\leq p-1$ and $m\geq0$ represents a p-adic number. Here is what I have so far: Set $\epsilon>0$ ...
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Showing the sequence of partial sums of a power series is Cauchy under the p-adic norm

I want to show that the sequence $e_n = \sum_{i = 0} \frac{x^i}{i!}$ is Cauchy with respect to the p-adic norm when $p > 2$ and $|x|_p < 1$. So far, I have the estimate $|e_n - e_m|_p = |\sum_{i ...
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'Smooth' p-adic analysis (perhaps via toposes)

There are sensible theories of analytic functions on non-Archimedean fields (rigid analytic spaces, Berkovich spaces), but these are modeled after complex analysis. I'm curious to what extent there ...
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Prove that $a \equiv b \mod{p^j} \iff |a-b|_{p}\leq p^{-j}$

Prove that $a \equiv b \mod{p^j} \iff |a-b|_{p}\leq p^{-j}$ Typically, when dealing with a congruence I go to the division statement. i.e $$a\equiv b\mod{p^j}\Rightarrow p^j|a-b \;\;\;(\star)$$ ...
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1answer
36 views

Reconstructing formal groups from the p-map, realizing p-maps from formal group

Suppose $F$ is a formal group over $\mathbb{Z}_p$. There are few trivial condition that $f \in \mathbb{Z}_p[[t]]$, the power series representing the $p$-map should satisfy: 1)$f \equiv g(t^p) mod p$ ...
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Integral closure of the p-adic integers in a finite extension of the p-adic numbers

In Cassels' article "Global Fields", he uses the term "ring of integers" and the notation $\mathcal{O}_K$, where $K$ is a field with a non-archimedean valuation, to denote the ring of elements $x \in ...
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91 views

Quotient of $\mathbb{Z}_p$ by the rational integers

Let $p$ be a prime and let $\mathbb{Z}_p$ denote the $p$-adic integers. One has a canonical inclusion of rings $$\mathbb{Z}_{(p)}\longrightarrow\mathbb{Z}_p$$ given by identifying the rational ...