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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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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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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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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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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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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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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85 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 ...
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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? ...
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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 ...
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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 ...
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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 ...
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158 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.
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107 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 ...
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$2$-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 ...
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37 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 ...
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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 ...
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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 ...
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88 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.
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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 ...
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54 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 ...
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62 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 ...
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134 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 ...
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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}), ...
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Equivalent Hensel's lemma?

Let $K$ be a complete field with nonarchimedian valuation $|\cdot|$ and $\mathcal{O}_K := \{x \in K: \; |x| \le 1\}$, $\mathfrak{m} := \{x \in K: \; |x| < 1\}$. I have seen two statements that do ...
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Every finite Galois extension of $\mathbb{Q}_p$ has a solvable Galois group.

I have found the statement as in the title of this post on wikipedia, however, there is no reference for its proof. How does one prove it?
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Can one show a beginning student how to use the $p$-adics to solve a problem?

I recently had a discussion about how to teach $p$-adic numbers to high school students. One person mentioned that they found it difficult to get used to $p$-adics because no one told them why the ...
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Computing the analytic $p$-adic $L$-function via modular symbols in MAGMA

I need to compute the analytic $p$-adic $L$-function of an elliptic curve at a prime $p$ via modular symbols using MAGMA. In ...
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$\mathbb{Q}_p\otimes_{\mathbb{Q}} \mathbb{Q}_q$ and $\mathbb{Z}_p\otimes_{\mathbb{Z}} \mathbb{Z}_q$

Let $p, q$ be prime numbers which may or may not be distinct. Let $\mathbb{Q}_p$ be the field of $p$-adic numbers. Let $\mathbb{Z}_p$ be the ring of $p$-adic integers. We define similarly ...
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GL$_2(\mathbb{Q}) Z_{\mathbb{R}}$ closed in GL$_2(\mathbb{A})$?

I am struggling with the following subgroups of GL$_2(\mathbb{A})$ where $\mathbb{A}$ is (the topological ring of) Adeles over $\mathbb{Q}$: $$G_\mathbb{Q} := \iota(\text{GL}_2(\mathbb{Q})) $$ where ...
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Let $p \equiv 2 \mod{3}$. For any $a \in Z$ such that $ p \nmid a$ , show that there exists $x \in \mathbb{Z}_p$ with $x^3 = a$.

Let $p \equiv 2 \mod{3}$. For any $a \in Z$ such that $ p \nmid a$ , show that there exists $x \in \mathbb{Z}_p$ with $x^3 = a$. I've tried using Hensel's lemma and the fact that if $p \equiv 2 ...
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Show that the curve $2Y^2 = X^4-17$ has points in every $\mathbb{Q}_p$

I've been asked to show that the curve $2Y^2 = X^4-17$ has points in every $\mathbb{Q}_p$ - I've managed to show that it is birationally equivalent to the curve $Y^2 = 2X^4 - 34$ (as suggested in the ...
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Question about the cyclotomic $\mathbb Z_p$-extension

Let $K$ be a number field and $K_{\infty}/K$ the cyclotomic $\mathbb Z_p$-extension of $K.$ My question is : How to prove that for any prime $\ell$ of $\mathbb Q$ distinct to $p$ does not decompose ...
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Dedekind-Like construction of p-adic numbers

Recently I've been studying p-adic numbers. I understand the idea of a cauchy completion of the rationals with respect to the metric defined by the norm $\vert\vert \cdot \vert \vert_p $. When I was ...
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Convergence over the $p$-adic numbers

so I'm a bit confused about convergence over the $p$-adic numbers - for example, I would argue that $\frac{1}{5^n}$ is not convergent over $\mathbb{Q}_5$ since $|\frac{1}{5^n}|_p = 5^n$ which is not ...
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$p$-adic approximation

I am asked to find $x\in\mathbb{Z}$ s.t. $|x^2+1|_5\leq5^{-4}$. I have a method of doing it but I'm not sure if it's right and whether my conclusion is correct either. It also seems exceedingly long ...
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Rational Number Form

I was reading Rational Points on Elliptic Curves by Silverman and Tate and they state: Every non-zero rational number may be uniquely written in the form $\frac{m}{n}p^v$, where $m,n$ are integers ...
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$p$-adic logarithm

For $q\in\mathbb{C}_p$ such that $|q|_p < 1$ show that there exists a unique logarithm $\log_q:\mathbb{C}_p^{*}\to\mathbb{C}_p$ with (i) $\log_q(q)=0$ (ii) $\forall x\in\mathbb{C}_p$, ...
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An application of Hensel's lemma to some different fields

I would like to prove the following. Let $p>2$ be a prime number, $\mathbb{Q}_{p}$ the field of p-adic numbers Let $u\in \mathbb{Z}_{p}^{\times}$ be a unit. (1)Prove that the following are ...
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$p$-adic valuation.

Let $\alpha_1,\alpha_2\in \mathbb Z_p$ such that $v_p(\alpha_1)<v_p(\alpha_2).$ How to prouve that $v_p(\alpha_2-\alpha_1)=v_p(\alpha_1)$ ? I think this is a stupid question but I'm really ...
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Nilpotent action on $p$-group

Let $A$ be a finite, abelian $p$-group and $\Gamma$ is a multiplicative topological group isomorphic with the additive group of $p$−adic integers $\mathbb Z_p.$ and let $\gamma_0$ a topological ...
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Writing $p$-groups using $p$-adics

Is it possible to write any finite abelian $p$-group as $\mathbb{Z}_p^n/\mbox{im }(A)$ for some $n\times n$ matrix $A$ over $\mathbb{Z}_p$? Here $\mathbb{Z}_p$ denotes the $p$-adic integers.
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Question about $p$-adic exponential

Let $p$ be a prime number, and $K$ a finite extension of $\mathbb Q_p$ and $S=p^N\mathcal O_K$ where $\mathcal O_K$ the ring of integers of $K.$ I know that for $N>>0$ enough large the $p$-adic ...
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Constructing the complex p-adic numbers

I'm reading through "$p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions" by Koblitz to learn about p-adic numbers. In chapter 3, he describes the construction of $\Omega$ (a.k.a. $\Omega_p$), the ...
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Topological isomorphism of projective system

Let $K_0 \subset K_1 \subset ...K_n\subset...$ be a tower of normal number fields, and put $G_n = \mathrm {Gal} (K_n/K_0).$ Define epimorphisms $\pi_{mn} :G_n\longrightarrow G_{m}$ for $m < n$ ...