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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Possible Quadratic extensions of $\mathbb{Q}_{2}$ are 1-1 with $\mathbb{Q}_{2}^{*}/(\mathbb{Q}_{2}^{*})^{2}$

Why do they have to be units? $\mathbb{Q}_{2}[\sqrt{2}]$ is a quadratic extension but $|2|_{2}=\frac{1}{2}\neq 1$. Where do I need them to be units below? ...
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A quadratic equation over $\mathbb{Q}_p$

Suppose we have the equation $x^2+x+1$ over the field $\mathbb{Q}_p$. is it possible to determine for what primes $p$ the equation has solutions? I tried to see whether this is related to what $p$ is ...
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relation between $p$-adic numbers and congruence relations modulo p

Set $p$ prime. Take $f_1, f_2 \in \mathbb{Z}$ with $f_1 f_2$ square free and $p \nmid f_1, f_2$ and $\alpha \in \mathbb{Q}_p$ such that $|\alpha|_p = 1$ (i.e. $\alpha$ is a $p$-adic unit). Show that ...
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What is the p-adic valuation of elements of $\mathbb{Q}_p$ not in $\mathbb{Q}$.

In Cassels book "Lectures on Elliptic Curves", he defines the $p$-adic integers as: $\quad \mathbb{Z}_p = \{\alpha \in \mathbb{Q}_p \mid |\alpha|_p \leq 1\}$ He latter states that the $p$-adic ...
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Problems that are made easy by using p-adic numbers

Does anybody know of elementary problems that can be be solved using the p-adics? Solutions are preferred.
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Open Ball under the p-adic Norm

I'm trying to figure how, if it's even possible, to draw an open ball using the p-adic norm. My definition of the p-adic norm I'm using is: $ \lvert x \rvert_p $ = $p^{-ord_px}$ if $x \neq 0$ and ...
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$2$-adic inverse of $3$

Performing long division, I calculated the $2$-adic inverse of $3$ to be $$1-2+4-8+...$$ Then I noticed that I could get the same result in a more sleek way by noting that $$\frac 1 3 = \frac ...
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Ramification and roots of unity in complete discrete valuation rings.

Let $\mathcal{O}$ be a complete discrete valuation ring with algebraically closed residue field $k$ of characteristic $p>0$. Let $\pi\in \mathcal{O}$ generate the maximal ideal and suppose ...
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Degree of closure of $\mathbb{Q}_p$

In order to prove that algebraic closure of $\mathbb{Q}_p$ is infinite, I took the polynomial $x^n-p$ with $n>1$ over $\mathbb{Q}_p$ to show that this eqaution has no solution for infinite cases to ...
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What does root of unity in $\mathbb{Z}_p$ look like?

Let $p$ be an odd prime. Then by Hensel's lemma it's clear that $\mathbb{Z}_p $ contains all $p-1$th root of unity which reduces to $1$, $2$, ... , $p-1$ in $\mathbb{F}_p$. My question is do we know ...
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What is $e, \pi, \ln 2,…$ etc in p adic?

What is $e, \pi, \ln 2,...$ etc in p adic? And how to flip digits of decimal points? Does p-adic have their own constants? 10 adic base.
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Open problems involving p-adic numbers

I am in my final year of my undergraduate degree, and I'm doing a project on p-adic numbers, and in particular, trying to find Galois groups of simple extensions of $\mathbb{Q}_p$ (this is a Galois ...
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$p$-part of cyclotomic character

Let $K$ be a number field, $\bar K$ a separable closure and $p$ a rational prime and assume that $p \not= char(K)$. Consider the extension $$K(\mu_{p^\infty}) \mid K$$ which is obtained by adjoining ...
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Number of field extensions of $\mathbb{Q}_p$

If I know the index $(\mathbb{Q}_p^{\times} : (\mathbb{Q}_p^{\times})^n)$ for some $n \in \mathbb{N}$, is it possible to know how many field extensions of $\mathbb{Q}_p$ of degree $n$ there are? This ...
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Embedding into $p$-adic complex numbers

As I'm reading notes about the Leopoldt conjecture, the following question came to my mind: Let $\mathbb{C}_p$ be the $p$-adic complex numbers, i.e. the completion of the algebraic closure of the ...
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The unramified quadratic extension of $\mathbb{Q}_2$

I know that there are $7$ field extensions of $\mathbb{Q}_2$ of degree $2$ (this follows from Hensel's lemma) and I think these are $$\mathbb{Q}_2(\sqrt{2}), \mathbb{Q}_2(\sqrt{3}), ...
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Equivalent Hensel's lemma?

Let $K$ be a complete field with nonarchimedian valuation $|\cdot|$ and $\mathcal{O}_K := \{x \in K: \; |x| \le 1\}$, $\mathfrak{m} := \{x \in K: \; |x| < 1\}$. I have seen two statements that do ...
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Every finite Galois extension of $\mathbb{Q}_p$ has a solvable Galois group.

I have found the statement as in the title of this post on wikipedia, however, there is no reference for its proof. How does one prove it?
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Can one show a beginning student how to use the $p$-adics to solve a problem?

I recently had a discussion about how to teach $p$-adic numbers to high school students. One person mentioned that they found it difficult to get used to $p$-adics because no one told them why the ...
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Computing the analytic $p$-adic $L$-function via modular symbols in MAGMA

I need to compute the analytic $p$-adic $L$-function of an elliptic curve at a prime $p$ via modular symbols using MAGMA. In ...
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$\mathbb{Q}_p\otimes_{\mathbb{Q}} \mathbb{Q}_q$ and $\mathbb{Z}_p\otimes_{\mathbb{Z}} \mathbb{Z}_q$

Let $p, q$ be prime numbers which may or may not be distinct. Let $\mathbb{Q}_p$ be the field of $p$-adic numbers. Let $\mathbb{Z}_p$ be the ring of $p$-adic integers. We define similarly ...
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GL$_2(\mathbb{Q}) Z_{\mathbb{R}}$ closed in GL$_2(\mathbb{A})$?

I am struggling with the following subgroups of GL$_2(\mathbb{A})$ where $\mathbb{A}$ is (the topological ring of) Adeles over $\mathbb{Q}$: $$G_\mathbb{Q} := \iota(\text{GL}_2(\mathbb{Q})) $$ where ...
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Let $p \equiv 2 \mod{3}$. For any $a \in Z$ such that $ p \nmid a$ , show that there exists $x \in \mathbb{Z}_p$ with $x^3 = a$.

Let $p \equiv 2 \mod{3}$. For any $a \in Z$ such that $ p \nmid a$ , show that there exists $x \in \mathbb{Z}_p$ with $x^3 = a$. I've tried using Hensel's lemma and the fact that if $p \equiv 2 ...
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Show that the curve $2Y^2 = X^4-17$ has points in every $\mathbb{Q}_p$

I've been asked to show that the curve $2Y^2 = X^4-17$ has points in every $\mathbb{Q}_p$ - I've managed to show that it is birationally equivalent to the curve $Y^2 = 2X^4 - 34$ (as suggested in the ...
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Question about the cyclotomic $\mathbb Z_p$-extension

Let $K$ be a number field and $K_{\infty}/K$ the cyclotomic $\mathbb Z_p$-extension of $K.$ My question is : How to prove that for any prime $\ell$ of $\mathbb Q$ distinct to $p$ does not decompose ...
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Dedekind-Like construction of p-adic numbers

Recently I've been studying p-adic numbers. I understand the idea of a cauchy completion of the rationals with respect to the metric defined by the norm $\vert\vert \cdot \vert \vert_p $. When I was ...
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Convergence over the $p$-adic numbers

so I'm a bit confused about convergence over the $p$-adic numbers - for example, I would argue that $\frac{1}{5^n}$ is not convergent over $\mathbb{Q}_5$ since $|\frac{1}{5^n}|_p = 5^n$ which is not ...
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$p$-adic approximation

I am asked to find $x\in\mathbb{Z}$ s.t. $|x^2+1|_5\leq5^{-4}$. I have a method of doing it but I'm not sure if it's right and whether my conclusion is correct either. It also seems exceedingly long ...
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Rational Number Form

I was reading Rational Points on Elliptic Curves by Silverman and Tate and they state: Every non-zero rational number may be uniquely written in the form $\frac{m}{n}p^v$, where $m,n$ are integers ...
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$p$-adic logarithm

For $q\in\mathbb{C}_p$ such that $|q|_p < 1$ show that there exists a unique logarithm $\log_q:\mathbb{C}_p^{*}\to\mathbb{C}_p$ with (i) $\log_q(q)=0$ (ii) $\forall x\in\mathbb{C}_p$, ...
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An application of Hensel's lemma to some different fields

I would like to prove the following. Let $p>2$ be a prime number, $\mathbb{Q}_{p}$ the field of p-adic numbers Let $u\in \mathbb{Z}_{p}^{\times}$ be a unit. (1)Prove that the following are ...
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$p$-adic valuation.

Let $\alpha_1,\alpha_2\in \mathbb Z_p$ such that $v_p(\alpha_1)<v_p(\alpha_2).$ How to prouve that $v_p(\alpha_2-\alpha_1)=v_p(\alpha_1)$ ? I think this is a stupid question but I'm really ...
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Nilpotent action on $p$-group

Let $A$ be a finite, abelian $p$-group and $\Gamma$ is a multiplicative topological group isomorphic with the additive group of $p$−adic integers $\mathbb Z_p.$ and let $\gamma_0$ a topological ...
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Writing $p$-groups using $p$-adics

Is it possible to write any finite abelian $p$-group as $\mathbb{Z}_p^n/\mbox{im }(A)$ for some $n\times n$ matrix $A$ over $\mathbb{Z}_p$? Here $\mathbb{Z}_p$ denotes the $p$-adic integers.
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Question about $p$-adic exponential

Let $p$ be a prime number, and $K$ a finite extension of $\mathbb Q_p$ and $S=p^N\mathcal O_K$ where $\mathcal O_K$ the ring of integers of $K.$ I know that for $N>>0$ enough large the $p$-adic ...
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Constructing the complex p-adic numbers

I'm reading through "$p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions" by Koblitz to learn about p-adic numbers. In chapter 3, he describes the construction of $\Omega$ (a.k.a. $\Omega_p$), the ...
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Topological isomorphism of projective system

Let $K_0 \subset K_1 \subset ...K_n\subset...$ be a tower of normal number fields, and put $G_n = \mathrm {Gal} (K_n/K_0).$ Define epimorphisms $\pi_{mn} :G_n\longrightarrow G_{m}$ for $m < n$ ...
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Embedding $\mathbb Q^c$ into $\mathbb Q^c_p$

Let $p$ be a prime number and $\mathbb Q_p$ the $p$-adic completion of $\mathbb Q$. Let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$. Is there an embedding $$j: \mathbb Q^c ...
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Definition of $\mathbb Q^c_p$

Let $p$ be a prime number and let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q $ in $\mathbb C$, i.e. the field of algebraic numbers. Is it possible at all to define the $p$-adic completion ...
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What do they mean by this: “the definition of product topo shows that this mapping is continuous”

Let $\Bbb{Z}_2$ be the $2$-adic integers. There's a bijection $\psi : \Bbb{Z}_2 \to C$, the Cantor set, defined by $\psi (\sum_{i \geq 0} a_i p^i) = \sum_{i \geq 0} \dfrac{2a_i}{3^{i+1}}$. The text ...
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How do you prove that $\Bbb{Z}_p$ is an integral domain?

Let $\Bbb{Z}_p$ be the $p$-adic integers given by formal series $\sum_{i\geq 0} a_i p^i$. I'm having trouble proving that it's an integral domain.
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Norm of $\mathbb{Q}_2(i)^\times$

I am trying to compute the norm group of $\mathbb{Q}_2(i)^\times$. If i'm not mistaken we have $\mathbb{Q}_2(i)^\times = (1+i)^\mathbb{Z}\mathbb{Z}_2[i]^\times$ so $N(\mathbb{Q}_2(i)^\times) = ...
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Group homomorphism $\mathbb{Z}_p \longrightarrow\mathbb{F}_p$

Let $p$ be a prime number , denote by $\mathbb{Z}_p$ the ring of $p-$adic integers. we can defined a group homomorphism $\mathbb{Z}_p \longrightarrow\mathbb{F}_p$ ?
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Intersection of p-adic integers

If we have two primes, $p$ and $p'$ and the $p,p'$-adic integers $\mathbb{Z}_p$ and $\mathbb{Z}_{p'}$. Then the integers inject into both of the rings, so we can say that $\mathbb{Z} \subset ...
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$Q_p(\zeta)$ where $\zeta$ is a $p$-th root of $1$.

I'm not looking for a full solution, only a hint please! Let $\zeta$ be a $p$-th root of unity in an algebraic closure of $Q_p$. Show that $Q_p(\zeta) = Q_p ((-p)^{\frac{1}{p-1}})$. Following a hint ...
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$v$-adic ring of integers of a number field

Let $K/\mathbb{Q}$ be a number field and $v$ a finite valuation of $K$. We can consider the completion $K_v$, which is a finite extension of $\mathbb{Q}_v$. We can define a "$v$-adic ring ...
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$p$-adic completion of $\mathbb{Z}[X]$ and $\mathbb{Z}[[X]]$.

Let $p$ be prime. The $p$-adic completion of $\mathbb{Z}$ is the ring $\mathbb{Z}_p$ of $p$-adic integers, and its elements can be thought of as power series in $p$. Is there a nice description of the ...
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Lines in a p-adic plane

The geometry of lines in $\mathbb{R}^2$ is fundamental to mathematics and likewise for lines in $\mathbb{C}^2$ since $\mathbb{C}^2 \cong \mathbb{R}^4$. But is there a good treatment of lines in ...
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Arithmetic in p-adic

I came to the problem saying that there exists a number $x\in \Bbb Z_7$ such that $x^2=2$ but there is no such $x$ in $\Bbb Z_5$. Could anyone give an explanation of this? How to actually find the ...
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Maps between spectral sequences

I am trying to understand a subtle point about how Theorem 2.2.5 is used in Kedlaya, Abbott, and Roe's "Bounding Picard numbers of surfaces using p-adic cohomology". Below I've tried to pose the ...