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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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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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: $\left|\sum_0^N (p-1)p^i+1 \right|_p ...
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$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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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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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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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? ...
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The equation $P(X,Y)$ has a solution in $\mathbb{Q}_p$

Proposition Let $P(X,Y) \in \mathbb{Z}[X,Y]$. The following propositions are equivalent: The equation $P(X,Y)$ has a solution in $\mathbb{Q}_p$. For each $n \geq 0$ the equation $P(X,Y)$ has a ...
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Idea of Hensel's Lemma

$$P(x, y)=0 \tag {1} \\ \text{ where }P(x, y) \in \mathbb{Q}[x, y]$$ How do we know that $(1)$ has a solution in a $\mathbb{Q}_p$ ? We will apply Hensel's Lemma. Idea: we begin from an element ...
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$K-$rational solution of the equation - Is $\mathbb{Q} \leq \mathbb{Q}_p$?

Let $P(x, y) \in \mathbb{Q}[x, y]$. We consider the equation $P(x, y)=0$. If $a, b \in \mathbb{Q}$ such that $P(a, b)=0$ then $(a, b) \in \mathbb{Q}^2$, is called a rational solution. If $K$ a ...
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How do we deduce that $\mathbb{Q}_p=\{ p^mu\mid u \in \mathbb{Z}_p^{\star}, m \in \mathbb{Z}\} \cup \{0\}$?

We know that each element $x$ of $\mathbb{Z} \setminus{\{0\}}$ has a unique representation of the form $x=p^m u\mid m \in \mathbb{N}_0, u \in \mathbb{Z}_p^{\star}$. $$\mathbb{Q}_p=\left \{ ...
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Diophantine equation not solvable in $\mathbb{Q}$, but in $\mathcal{O}_p$

I'm trying to think of an example of a diophantine equation which can be solved in $ \mathcal{O}_p$ (meaning it can be solved $\mod p^k$ for all $ k $) for all prime $ p $'s, but not in $\mathbb{Q}$ ...
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Taylor expansion in $p$-adic integers

Let $f \in Z_p[X]$, then for $ x, y \in Z_p$, $\exists a \in Z_p$ s.t. $f(y)=f(x)+(y-x)f'(x)+(y-x)^2a$. Why is Taylor formula applicable to polynomial in $p$-adic integers $Z_p$? What condition ...
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Show an $R$-module is a direct limit

This is a scenario I've encountered in my class on $p$-adic L functions. Let $G$ be a profinite group which is the inverse limit of a system $(G_i, f_{ij})$ of discrete finite topological groups. ...
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Efficient Method/Algorithm to Compute the P-adic Ordinal of a Positive Integer

I am currently writing my master's thesis at Cal Poly Pomona, and am currently investigating the ruler sequence for a prime base. The ruler sequence for base $2$ is : ...
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Why does it suffice to show it for positive integers?

I am looking at the proof of the product formula theorem: For each $x \in \mathbb{Q}$, it holds $$\prod_{p \leq \infty} |x|_p=1$$ The proof starts by this: It is enough to show it for ...
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Herbrand Quotient: Formula for $|K^*/(K^*)^n|$

NB: Please restrict answers to hints and not solutions. Problem: Use the theory of the Herbrand quotient $q(A)=H^{0}(A)/H^{1}(A)$ to show that, if $K$ is a finite extension of $\mathbb{Q}_p$, and ...
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Explicitly computing uniformisers of local fields

Consider the field tower $L/K'/K$ where $L=\mathbb{Q_3}(\xi,2^{1/3})$, $K'=\mathbb{Q_3}(\xi) $ and $K=\mathbb{Q_3}$. Here, $\xi$ is a primitive cube root of unity, and $\mathbb{Q_3}$ is the 3-adics. ...
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Finite index subgroup $G$ of $\mathbb{Z}_p$ is open.

Suppose $[\mathbb{Z}_p:G] = n <\infty$. Write $n = p^km$ with $p\nmid m$. The idea is to show that $p^k\mathbb{Z}_p = n\mathbb{Z}_p \subseteq G$, after which I am done, since for any $x\in G$ we ...
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For $p\ge3$ there is no extension of $\mathbb Q_p$ with Galois group $S_4$

I'm trying to show that if $p\ge3$ is prime, then There is no extension $K$ of the field of $p$-adic numbers $\mathbb Q_p$ with Galois group $S_4$. I know that $K$ must have a subextension ...
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$0$ and $p^n \mathbb{Z}_p$ are the only ideals of $\mathbb{Z}_p$

I am looking at the proof of the sentence: $\mathbb{Z}_p$ contains only the ideals $0$ and $p^n \mathbb{Z}_p$ for $n \in \mathbb{N}_0$. It holds that $\bigcap_{n \in \mathbb{N}_0 p^n \mathbb{Z}_p}=0$ ...
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Is $123456788910111121314\cdots$ a $p$-adic integer?

On the back of this question comes the natural question of whether the string $$1234567891011121314\!\cdots$$ is even a number at all. While that sort of question is vague, given the lack of generic ...
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Why is this function an embedding?

We have the canonical function $\epsilon_p: \mathbb{Z} \to \mathbb{Z}_p, x \mapsto (\overline{x})_{k \in \mathbb{N}_0}=(\overline{x}, \overline{x}, \overline{x}, \dots )$. The function $\epsilon_p: ...