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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$3 X^3 + 4 Y^3 + 5 Z^3$ 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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50 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 ...
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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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19 views

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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22 views

Showing Zp is isomorphic to the completion of Z with the p-adic norm using Cauchy sequences

Following on from James' question, here: Showing Zp is isomorphic to the completion of Z, when Z is considered as a p-adic metric space I understand that every element a = a1,a2,a3,... of Zp can thus ...
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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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22 views

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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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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48 views

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
23 views

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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20 views

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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12 views

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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43 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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37 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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41 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 ...
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Haar measure and p-adics

While studying the theorem of existence and uniqueness of the Haar measure, I was asked to find the unique linear functional $E : C_{\mathbb{R}}(\mathbb{Z}_P) \longrightarrow \mathbb{R}$ that ...
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23 views

Given two extensions of $\left| \cdot \right|_p$ to $\Bbb{C}$, do the subsets of elements of absolute value $1$ coincide?

It is known that $\Bbb{C}$ is isomorphic as a field to $\Bbb{C}_p$, the completion of $\bar{\Bbb{Q}}_p$ with respect to $\left|\cdot\right|_p$. Clearly, given two such isomorphisms $\varphi_1$ and ...
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Examples of an unbounded measurable subset of finite measure of the p-adic number field

Let $\mathbb{Q}_p$ be the p-adic number field, $\mathbb{Z}_p$ its ring of integers. A subset of $\mathbb{Q}_p$ is called bounded if it is contained in a compact subset. Let $\mu$ be the Haar measure ...
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Explicit construction of Haar mesure on the p-adic number field

Let $\mathbb{Q}_p$ be the p-adic number field, $\mathbb{Z}_p$ its ring of integers. Let $\mathcal B$ be the smallest $\sigma$-algebra containing all the open subsets of $\mathbb{Q}_p$. Can we prove ...
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Weil group of non archimedian field

Let $F$ be a finite extension of $\Bbb{Q}_p$. Let $W_F$ be the Weil group of $F$. Let $I_F$ be the inertia group of $F$. Let $\phi$ be an element of the weil group of $F$ which does not belong to the ...
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Is $\sum_{n = 1}^{\infty} v_p(n) 2^{-n+1}$ a rational number?

Fix $p \in \Bbb{Z}$ a prime number and let $v_p$ be the usual $p$-adic valuation on $\Bbb{Q}$. I would like to know if $$ \sum_{n = 1}^{\infty} \frac{v_p(n)}{2^{n-1}} $$ is a rational number. I ...
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11 views

Additive inverse of elements in the p-adic numbers $\mathbb Q_p$

I am trying to find out if $-a = -1\cdot a$ where $-a$ is the additive inverse of $a=a_{-m}\cdot p^{-m}+\cdots + a_{-1}p^{-1} + a_0 + a_1\cdot p +\cdots$ in $\mathbb Q_p$ and $-1= (p-1)+(p-1)\cdot ...
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29 views

Algorithm to determine if a given polynomial over $\mathbb{Q}$ is irreducible over the $p$-adic number field

Let $\mathbb{Q}_p$ be the $p$-adic number field. Let $f(x) \in \mathbb{Q}[x]$ be a polynomial of degree $\ge 1$. Is there an algorithm to determine whether $f(x)$ is irreducible over ...
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62 views

$p$-adic numbers and residually finiteness

Let $J_p$ be the additive group of $p$-adic integers and $Q_p$ be the additive group of $p$-adic numbers. I know that $J_p$ is residually finite. Is $Q_p$ residually finite? Definition A group $G$ ...
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$\mathbb{Z}_p$ as a module over $\mathbb{Z}_{(p)}$

I denote by $\mathbb{Z}_{(p)}$ the localization at a prime p, and by $\mathbb{Z}_p$ the p-adic integers. Question: what is the structure of $\mathbb{Z}_p$ as $\mathbb{Z}_{(p)}$- module? For example ...
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Method of finding a p-adic expansion to a rational number

Could someone go though the method of finding a p-adic expansion of say $-\frac{1}{6}$ in $\mathbb{Z}_7?$
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72 views

2-adic expansion of (2/3)

I have been asked in an assignment to compute the 2-adic expansion of (2/3). It just doesn't seem to work for me though. In our definition of a p-adic expansion we have $x= \sum_{n=0}^{\infty}a_np^n$ ...
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1answer
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p-adic expansions and reciprocal

Trying to get my head around p-adics,as i learn some more advanced number theory techniques but im stuck on an exercise. If we let $\alpha = a_0 + a_1p + a_2p^2 + \dots = \sum_{n=0}^\infty a_np^n$ be ...
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A question about the additive group of the $p$-adic integers

Let $J_p$ be the additive group of the $p$-adic integers. I know that it is torsion-free. I'm not pretty confortable with $p$-adic. Is it possible to find a direct sum of infinitely many cyclic ...
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57 views

The maximal unramified extension of a local field may not be complete

While reading my notes of a course in local class field theory, I arrived to a remark where it is said that given a complete discrete valuation field $K$, its maximal unramified extension $$K^{ur}= ...
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38 views

Inverse limit and union of $\mathbb{Z} / p^{n}\mathbb{Z} $

Let $p$ be a prime number and let the natural embeddings $\mathbb{Z}/p\mathbb{Z} \subset \mathbb{Z}/p^{2}\mathbb{Z} \subset \dots \subset \mathbb{Z}/p^n\mathbb{Z} \subset \dots $ Questions: Does ...
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Does $\sum n $ converge p-adically?

Does $\sum n $ converge p-adically, I have worked out $v_p(n) \leqslant log(n)/log(p) $ not sure how to conclude from this I want to prove this using the result that it converges p-adically iff ...
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30 views

Calculating p-adic valuation $v_p(n)$, using basic properties

Calculating p-adic valuation $v_p(n)$ I'm not confident with the properties of $v_p(n)$ Where $v_p(n) = $ the biggest integer $e$ such that $p^e$ divides $n$, if $n\not=0$, and $+\infty$ if $n=0$. ...
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Norm group of non-Abelian Galois extension of p-adic numbers

Let $K=\mathbb Q_p(\xi_p,\alpha)$ be an extension of $\mathbb Q_p$, where $\xi_p$ is a primitive $p$-th root of the unite and $\alpha$ is a root of $X^p=p$. Now $K/\mathbb Q_p$ is a Galois entension ...
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the different of finite extension of p-adic numbers

Let $F=\mathbb Q_p$ be the field of p-aidc numbers. Let $\xi_n$ be a $n$-th primitive root of unity. Now consider the finite extension of fields $K/F$ where $K=\mathbb Q_p(\xi_n)$. I want to find the ...
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75 views

How to find a p-adic expansion?

I've been reading about p-adic numbers recently and came across a question that asks to find the $5$-adic expansion of -3. I've been unable to find any similar examples so I can see how to work my ...
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Regarding constructing the $I$-adic completion of a ring

Let $R$ be a commutative ring and let $I \subset R$ be an ideal. For $n \geq m$, let $\varphi_{m,n}$ denote the canonical ring homomorphism $R / I^n \to R / I^m$. Let $J = \cap_n I^n$. Then $R / J$ ...
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Lattices in $\mathbb{Q}_p^n$ with the same stabilizer

Consider the action of $GL_n(\mathbb{Q}_p)$ on $\mathbb{Q}_p^n$, and let $T$ be the diagonal torus. Let $\Lambda$ and $\Lambda'$ be full-rank sublattices ($\mathbb{Z}_p$-submodules of rank $n$) such ...
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What is meant by the form of a polynomial in $A_n$ deduced from a polynomial $f$ over $\mathbb{Z}_p$?

I am reading Serre's A Course in Arithmetic and am having trouble understanding what he means by a polynomial deduced from a polynomial over $\mathbb{Z}_p$. Specifically Serre writes, ...
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Iwasawa module: $O[[G]]$ is noetherian where $G$ is a compact $p$ adic Lie group

Let $G$ be a compact $p$ adic Lie group. Let $E$ be a finite extension of $\Bbb{Q}_p$ with ring of integers $O$. Then how to show that $O[[G]]$ is noetherian. I was reading the article here and on ...
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For which $a>0$ does the equation $x^2+y^2+z^2=a$ have a solution in $\mathbb{Q}_2$?

We want to check for which $a>0$ we have that the equation $x^2+y^2+z^2=a$ has a solution in $\mathbb{Q}_2$. $x^2+y^2+z^2=a$ has a solution in $\mathbb{Q}_2$ for $x \in \mathbb{Z}_2^{\star}, y ...
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$P$-adic Numbers Least Close to One

If I understand the definition of $p$-adic numbers, then the numbers that are $2$-adically least close to one are $3, 7, 11, \ldots$ because they are divisible by $2^1$. Do the two-adic numbers, ...
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54 views

Show that $-1=\sum_{0}^{\infty} (p-1)p^i$ in $\mathbf{Q}_p$

To show that in the field $\mathbb{Q}_p$, where $p$ is a prime, it holds that: $$-1=\sum_{0}^{\infty} (p-1)p^i$$ I did the following: It suffices to show that: $\left|\sum_0^N (p-1)p^i+1 \right|_p ...
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$f(x)=x^2-a \in \mathbb{Z}[x]$ - show propositions

Let $f(x)=x^2-a \in \mathbb{Z}[x]$. $$p \in \mathbb{P}, p \neq 2, p^2 \nmid a$$ The equation $f(x)=0$ If $p \mid a $, the equation has no solution in $\mathbb{Q}_p$ Let $p \nmid a$. The ...
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1answer
58 views

pth root of unity in $p$-adic field

It is well known that $\mathbb{Q}_p(\mu_n)$ is a totally ramified extension of degree $(p-1)p^n$ if $\mu_n$ is a primitive $p^n$th root of unity. However how true is this statement for a finite ...
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1answer
46 views

Question about the proof of Hensel's Lemma

Hensel's Lemma: Let $F(x)=\alpha_0 + \alpha_1 x+ \dots + \alpha_n x^n \in \mathbb{Z}_p[x]$. We suppose that there is a $p$-adic $a_1 \in \mathbb{Z}_p$ such that $$F(a_1) \equiv 0 \mod p ...
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There is a primitive $m^{th}$ root of unity in $\mathbb{Q}_p$ $\Leftrightarrow m \mid (p-1)$

After Hensel's Lemma there is the following proposition in my notes: If $p$ is a prime and $m \in \mathbb{N}$ then there is a primitive $m^{th}$ root of unity in $\mathbb{Q}_p$ $\Leftrightarrow m ...
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30 views

The definition of p-adic numbers

Are 3, 5, 7, 11, 13, 15 p-adically close to one because their differences with one are divisible by two, which is two to the first power? Possibly they are also equally distant from or close to one? ...