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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$p$-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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Question on p-adic norms and metric spaces

I was given this assignment question: I am given two distinct primes $ p \neq q $ and I am asked to produce first of all a set closed with respect to p-adic norm but not closed under q-adic norm. The ...
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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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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 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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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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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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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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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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38 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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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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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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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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$p$-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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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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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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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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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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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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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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1answer
49 views

$f(x)=x^2-a \in \mathbb{Z}[x]$ - 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
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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 ...