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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Proof that an archimedean absolute value on $\mathbb Q$ is equivalent to $|\ |_\infty$

I have included a bolded comment in a step in the part of Gouvea's proof of Ostrowski's theorem where he shows that an archimedean absolute value on $\mathbb Q$ is equivalent to $|\ |_\infty$ (the ...
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Squares in $\mathbb Z_p$

Let $p\neq 2$, then I want to understand that every element in $\mathbb Z_p$ (p-adic integers) is a square. For the prove one must see that $2$ is invertible in $\mathbb Z_p$. But $2$ is the element ...
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Roadmap to $p$-adic numbers: where a self-learner should look for references

TL;DR at the end of the question. I’m currently trying to learn as much as possible about p-adic numbers. I’m not sure what is the most fascinating part of the theory, but the use of the adjective ...
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Fermats little theorem for p-adic integers

Just a short question: Is Fermats little theorem applicable in the p-adic integers $\mathbb Z_p$? If yes, why?
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Structure of the group $\{1+p\mathbb Z_p \}$

In preparation for algebraic number theory I am reading Serre : A course in Arithmetic. I stuck in understanding a proof (p.17): Notation: $U_n=1+p^n\mathbb Z_p$ Actually there are many things ...
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The group $(1+p\mathbb Z_p)/(1+p^{n}\mathbb Z_p)$

I want know some information about the group \begin{equation*} \frac{(1+p\mathbb Z_p)}{(1+p^{n}\mathbb Z_p)} \end{equation*} (the Quotient group). What is the order of this group? I guess $p^{n-1}$ ...
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Which properties determine the uniqueness of the local Artin map?

Any abelian extension of local fields can be realized as the completion of a global abelian extension. So let $L/K$ be abelian, $w/v$ an extension of places. From the global Artin map on ideles we ...
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55 views

In $\mathbb Q_p$, proving every open ball is the disjoint union of more than one open ball

I'm reading the Foundations chapter of Gouvea's p-adic Numbers: An Introduction, and I'm trying to solve the following problem he poses to the reader: Take the $p-$adic absolute value on $\mathbb ...
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Why is $m \infty$ the conductor of $K = \mathbb{Q}(\zeta_m)/\mathbb{Q}$?

Wouldn't this be saying that for all $p$ dividing $m$, $1 + p^{\operatorname{ord}_p(m)} \mathbb{Z}_p$ is contained in the group of local norms $N_{\mathfrak p/p}(K_{\mathfrak p})$, where $\mathfrak p$ ...
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Show that $U_i \rightarrow U_{i+e}, x \mapsto x^p$ is an isomorphism

Show that $U_i \rightarrow U_{i+e}, x \mapsto x^p$ is an isomorphism? Let $K$ be a finite extension of $\mathbb{Q}_p$ with uniformizer $\pi$, prime $\mathfrak p$, and ramification index $e = ...
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Hensel's lemma & $p$-adic polynomial roots

I want to determine the number of roots of $f(X) = X^3-5X+20$ in $\mathbb{Z}_p$ using Hensel's lemma (lemma is on the bottom). Unfortunately I am not very well trained to solve this. Take for example ...
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Roots of $X^2-z$ in $2$-adics [duplicate]

Using Hensel's lemma is it true that $X^2-z$, for $z\in \mathbf Z$ any integer have no roots in 2-adics ? Hensel's lemma only shows, if a root of a polynomial can be lifted to a root in $\mathbf ...
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33 views

How to show that $X^2-2$ has a solution in $\mathbb Q_7$?

How to show that $X^2-2$ has a solution in $\mathbb Q_7$ ? Here, in example $5.3$ it is written that, there are $2$ solutions, but isn't justified why, do I have to check every coefficient or is ...
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35 views

Showing $\mathbf Z_2/8\mathbf Z_2\simeq\mathbf Z/8\mathbf Z$

Showing $\mathbf Z_2/8\mathbf Z_2\simeq\mathbf Z/8\mathbf Z$ $2$-adic integers are of the form $a_0+a_12+a_22^2+a_32^3+a_42^4\dots$ does modulo $8\mathbf Z$ means that, the only remaining part ...
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Solution of $8x^2+x+1=0\in \mathbb{Q}_2$.

We are asked to determine whether the equation $f(x)=8x^2+ x+ 1$ has a root in $\mathbb{Q}_2$. Now, I immediately think to apply Hensel's Lemma, with $\alpha=1$, by which I mean $f(\alpha)\equiv 0 ...
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Squares in $\mathbb Q_p$ are $p^{2n}\alpha$

If $p$ is an odd prime, then the squares in the field of p-adic numbers $\mathbb Q_p$ are the elements are $0$ or of the form $p^{2n}\alpha$, $n\in\mathbb Z$ and $\alpha\in\mathbb Z_p^{\times}$ ...
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A extending the p-adic valuation to a quadratic extension of $\mathbb{Q}_p$

I'm trying to solve the following problem. Prove that, if $d \in \mathbb{Z}_p$ is non-square, then $|a + b \sqrt{d}|p = |a^2 − b^2d|^{1/2}_p$ , for any $a, b \in \mathbb{Q}p$, defines a ...
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Questions about p-adic numbers

I got two questions about $p$-adic numbers: I often read that the field $\mathbb Q_p$ is much different than the field $\mathbb R$. An element of $\mathbb Q_p$ is of the form ...
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Topology of $\Bbb{Q}_p$

Let $a\in \Bbb{Q}_p$. Is $ a+p^x\Bbb{Z}_p$ an open set around $a$ in the topology of $\Bbb{Q}_p$. Here $x \in \Bbb{Z}$. Also I have another question. Is $\mathbb{Z}_p$ open in $\Bbb{Q}_p$?
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Does any continuous map between two Banach spaces over a nonarchimedian field is a closed map?

Does any continuous map between two Banach spaces is a closed map? It seems to me that it is true. Let $f:V \rightarrow W$ be a map, where $V$, $W$ are (infinite dimensional) Banach space over a a ...
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When is $x^2 - 75 y^2 = 0$ in $\mathbb{Z}_p$ solvable?

Exercise: For which prime numbers does the equation $x^2 - 75 y^2 = 0$ have non-trivial solution in the $p$-adic integers $\mathbb{Z}_p$? For $p\neq 5$, the non-trivial solvability of the ...
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Explicit description of $\Bbb Q_p \cap \bar{\Bbb Q}$

Note that we can embed $\Bbb Q_p$ into $\Bbb C$, as it is discussed here. But as far as I understand, this embedding sends the power series to transcendental elements, so we can't certainly embed ...
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Polynomial with a prime number as a root

Is it possible to prove that this equation is false: $$ \sum_{i=0}^n a_i p^i = 0 $$ with following conditions: $a_i \in [-1;1]$; [Might $a\in\{-1,1\}$ have been intended here?] $p$ is a prime ...
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Any finite index subgroup of $\mathbb Z_p$ is open [duplicate]

I'm trying to show that every finite index subgroup $H$ of $\mathbb Z_p$ is open. Since $H$ has finite index, it is equivalent (and perhaps easier) to show that it is closed. But I've tried showing ...
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1answer
24 views

non-archimedean absolute value (Ostrowski's theorem)

I'm reading the proof of Ostrowski's theorem in Gouvea's book on p-adic numbers and there is one step that I don't understand. Let $|\cdot|$ be a non-archimedean absolute value and $n=rp+s$ where ...
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27 views

“p-adic absolute value” in polynomial ring

I'm working with Gouvea's book on p-adic numbers. In problem 34 I'm asked to give a "p-adic" (I put it in quotes as in my understanding its just a p-adic-like) valuation and absolute value for an ...
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$p$-adic integers.

I have some problems regarding the definition of the $p$-adic integers. I've the following definition: $\mathbb{Z}_{p} = \lbrace \sum_{i=0}^{\infty}a_{i}p^{i}\vert a_{i}\in\mathbb{Z} \rbrace = ...
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Online calculator for $ p $-adic valuations and absolute values.

Does anyone know a website where I can enter a prime base and a rational and then get the $ p $-adic valuation and the $ p $-adic absolute value? For sure I know how to do it by hand, but I want to ...
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Relationship Between Ring of Integers of a Number Field to P-adic integers

Suppose we have a number field $K$ with ring of integers $\mathcal O_K$. Let $\frak p$ be a prime ideal in $K$ lying over $p\in \mathbb Q$. Then, using the $\frak p$-adic norm, we may define the ...
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Are p-adic numbers real or not? [closed]

Are p-adic numbers real numbers?Why or why not?I came across the idea that it is not a real number from-Is 0.9999... equal to -1? (last comment)
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Extension of the duality of the space of distributions over $X$ locally profinite space

I have two questions on a definition that appears in the book "Répresentations des groupes réductifs $p$-adiques" by David Renard (http://www.math.polytechnique.fr/~renard/Padic.pdf). Let $X$ be a ...
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Is the sequence $(v_p(n))$ of $p$-adic valuations of positive integers the fixed point of a morphism, for every prime $p$?

Fix a prime number $p$ and consider the sequence $\mathbf{v}_p = (v_p(n))_{n \geq 1}$, where $v_p$ is the usual $p$-adic valuation, i.e. $v_p(n) = a$ iff $p^a \parallel n$. While browsing the OEIS I ...
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$p$-adics with the least upper-bound property

It is well-known (I believe?) that the $p$-adics do not admit an ordering in the 'usual sense', the "usual sense" being a total order that is compatible with the field operations. I do not want to ...
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Closed subgroups of $\mathbb Q_p$

Let $\mathbb Q_p$ be the field of $p$-adic numbers, $\mathbb Z_p$ the ring of $p$-adic integers. Is there a closed subgroup of $\mathbb Q_p$ other than the following list? 1) 0 2) ...
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$3 X^3 + 4 Y^3 + 5 Z^3$ has roots in all $\mathbb{Q}_p$ and $\mathbb{R}$ but not in $\mathbb{Q}$

This is an exercise in my textbook in a chapter about the Hasse-Minkowski-theorem: Show that the polynomial $3 X^3 + 4 Y^3 + 5 Z^3$ has a non-trivial root in $\mathbb{R}$ and all $\mathbb{Q}_p$. ...
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Why is $\mathbb{Q}_\infty = \mathbb{R}$?

Why, in the context of p-adic numbers, do we have the convention $$\mathbb{Q}_\infty = \mathbb{R} \quad$$ ? It must have something to do with the generalization of the Legendre-symbol for ...
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What is the normal core of $GL_2(\mathbb{Z}_p)$ in $GL_2(\mathbb{Q}_p)$?

What is the normal core of $GL_2(\mathbb{Z}_p)$ in $GL_2(\mathbb{Q}_p)$? (the normal core of $H$ in $G$ is the largest subgroup of $H$ which is normal in $G$. it is the intersection of all ...
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Power series in p-adic integers

How can we show that for $x \in \mathbb{Z}_p$, $\log_p(1+x)$ converges in $\mathbb{Z}_p$ when $|x|_p < 1$? To clarify, $\log_p(1+x)$ is the power series: $$\sum_{n=1}^\infty ...
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Showing Zp is isomorphic to the completion of Z, when Z is considered as a p-adic metric space

Question: Show that Zp is isomorphic to the p-adic completion of Z; that is, the completion of Z when Z is considered a metric space via the p-adic metric. I'm stuck. If we take an element a in Zp, ...
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The “$GL_2(\mathbb{Z}[1/p])$-part” of an element in $GL_2(\mathbb{Q}_p)$

Is there a group homomorphism $\varphi:GL_2(\mathbb{Q}_p)\rightarrow GL_2(\mathbb{Z}[1/p])$ such that for every $x\in GL_2(\mathbb{Q}_p)$ we have $x^{-1}\cdot\varphi(x)\in GL_2(\mathbb{Z}_p)$? My ...
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Finite extnsion of local field.

Let $F$ be a finite extension of $\Bbb{Q}_p$. Let K be tamely ramified extension of $F$ Containing the maximal unramified extension. Let $P$ denote the residue field of the corresponding tamely ...
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Is 0.9999… equal to -1?

I read the following in a wikipedia article about 0.9999...: A third derivation was invented by a seventh-grader who was doubtful over her teacher's limiting argument that $0.999... = 1$ but was ...
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103 views

Is the algebraic closure of $\mathbb Q$ in the field $\mathbb Q_5$ a number field?

Is the algebraic closure of $\mathbb Q$ in the field $\mathbb Q_5$ of $5$-adic numbers a number field, if yes what is the degree ? To be honest I don't understand the question, what does it mean ...
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Why is $\frac{1}{3}=0.2\overline{31}$ in 5-adic expansion?

Why is $\frac{1}{3}=0.2\overline{31}$ in 5-adic expansion ? I get: ...
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1answer
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Convergence of the Riemann zeta function in $\mathbb Q_p$

Does the Riemann zeta function without p-Euler factor i.e. $\prod\limits_{\text{prime }q \not= p}\frac{1}{1-q^{-1}}$ converges in $\mathbb Q_p$?
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equivalent hensel lemma in the case of the rings Z/p*l Z

does exist an equivalent of the hensel lemma which permits the passage from the unity roots of finite field s into roots of unity in characteristic 0 ; in the case of the rings Z/p*l Z
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Character sums over witt rings of finite length

please I want to know if there is somme references for the studying the characters sums on the group of the Witt vectors of finite length
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52 views

$\mathbb{Q}_p$ not isomorphic to $\mathbb{Q}_q$ for $p\neq q$

I have to show that for $p, q$ prime and $p \neq q$ the fields $\mathbb{Q}_p$ and $\mathbb{Q}_q$ are not isomorphic. I have tried a proof but it seems far too complicated. What's the best way to ...
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1answer
39 views

How to interpret this formula for the Hilbert symbol?

I have a formula for the special case $p=2$ for the Hilbert symbol: \begin{equation*} \left( \frac{u 2^n , v 2^m}{2} \right) = ...
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47 views

For which $p$ is a number a square in $\mathbb{Q}_p$?

I have some numbers $r \in \{-1, 2, \frac{4}{5}, \ldots\}$ and have to find those primes $p$ for which $r$ is a square in $\mathbb{Q}_p$, i. e. is a solution of the equation $X^2 = r$ in ...