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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Reference for the p-adic numbers

Can anyone give me a reference (book or a paper) that introduces the p-adic numbers and their important properties? Also, I would love if that reference contained some not to advanced applications of ...
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Error in or another way of calculating $\frac{1}{2} \in \mathbb{Q}_3$

I want to find the $p$-adic expansion of $\frac{1}{2} \in \mathbb{Q}_3$. I begin with noting that $- \frac{1}{2} = \frac{1}{1-3} = \sum_{n=0}^{\infty} 3^n$. Therefore, $\frac{1}{2} = 1 - \frac{1}{2} = ...
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Understanding $p$-adic fields

OK, I'm completely lost on this. Define the $p$-adic integers $\mathbb{Z}_p$ as the projective limit $$\lim_{\leftarrow} \mathbb{Z}/p^n \mathbb{Z}.$$ So, if $a \in \mathbb{Z}_p$, then $a$ can be ...
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1answer
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Proof that $\mathbb Q_p$ is unique up to unique isomorphism preserving the absolute values

On pages 58-59 of Gouvea's $p-$adic Numbers: An Introduction, he gives the following proof that the field $\mathbb Q_p$, constructed using equivalence classes of Cauchy sequences, is unique up to ...
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Primitive $p^n$-th root of unity in $\bar{\mathbb{Q}}_p$.

I am trying to solve the following exercise in Koblitz's "$p$-adic Numbers, $p$-adic analysis, and Zeta-Functions". Let $p$ be a prime. Let $a$ be a primitive $p^n$-th root of unity in $\bar{\mathbb{...
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Solving equations in $\mathbb{Z}_3$ with Hensel's Lemma

Further to the post here, I'm trying to find the $n \in \mathbb{Z}$ such that there is a solution to the equation $$ x^3 +3x+y^3+3y=n$$ in $\mathbb{Z}_3$. Now, I've been able to show that in the ...
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Valuation of a particular element

I am tying to compute the valuation of a particular element of $\mathbb{Q}_p$. I am trying to compute $\operatorname{val}_p(P)$ where $P=\frac{\log(1+p^2)}{\log(1+p)}$ and $\log$ is the $p$-adic $\...
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Inverse limit of the following system.

Let $\mathcal{O}_L = \mathbb{Z}[\sqrt{-5}]$ be the number ring of $L=\mathbb{Q}[\sqrt{-5}]$, and $\mathfrak{p} =(2, 1+\sqrt{-5})$ a prime (and maximal ideal) in $\mathcal{O}_L$. What is the inverse (...
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$p$-Adic complex numbers, expansion into series

For any field of $p$-adic numbers $\mathbb{Q}_p$ there is a unique, up to an isomorphism, extension $\mathbb{C}_p$ that is both algebraically and metrically complete, the field of $p$-adic complex ...
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A slightly stranger Hensel's Lemma

I'm trying to understand the solution to this problem. It came up when doing some revision. It is essentially to show that the conclusion of Hensel's Lemma holds if we have take a valuation ring $R$ ...
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$p$(ain)-adic number sequence

I am trying to figure out how $p$-adic numbers work and currently am having trouble wrapping my head around how they work, so I made a pun! HAH! Jokes aside, I am working on this question Show ...
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Show an intersection of Galois groups is trivial

Let $L/K$ be a finite abelian extension of number fields, and for an extension of places $w/v$ consider the local Artin map $\Phi: K_v^{\ast} \rightarrow Gal(L_w/K_v)$, defined via the global Artin ...
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Is the $\mathbb{Q}_2$- Space$ (\mathbb{Q}_2[\zeta], trace(cxy))$ hyperbolic?

I am working at a Problem for some time and it comes down to the question: Let $K:=\mathbb{Q}_2[\zeta]$ be a cyclotomic extention of the dyadic field $\mathbb{Q}_2$. For any $c \in \mathbb{Q}_2[\zeta+...
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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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2answers
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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 “p-...
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2answers
159 views

Fermat's little theorem for p-adic integers

Just a short question: Is Fermat's 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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156 views

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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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 = 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 $\mathbf Z_p$...
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1answer
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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 is $...
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1answer
61 views

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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1answer
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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 $\alpha\in\mathbb Z_p^{\times}$ such ...
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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 non-...
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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 $\sum_{i=-k}^{...
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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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1answer
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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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1answer
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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 equation $...
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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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2answers
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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
54 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 $r,p,...
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1answer
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“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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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 $p$-...
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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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1answer
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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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1answer
80 views

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) $p^n\...
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1answer
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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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1answer
58 views

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 $\mathbb{R}$...
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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 $G$-...
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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 \frac{(-1)^{n+1}x^n}{n}...
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1answer
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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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29 views

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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1answer
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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 ...