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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infinitely $p$-divisible elements in $A\otimes \mathbb{Z}_p$

Let $A$ be a (possibly non-finitely generated) torsion-free abelian group. Suppose that $A$ contains no infinitely $p$-divisible elements, then does the same hold for $A\otimes \mathbb Z_p$, where $\...
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Where do these p-adic identities come from?

I was reading this article (http://www.asiapacific-mathnews.com/03/0304/0001_0006.pdf) to see some applications of $p$-adic numbers outside mathematics, and came across these two identities: $\sum_{n=...
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Is torsion of a topological module closed?

I was asking to myself the following question. Consider a $p$-adically complete and separated topological algebra $R$ over $\mathbb{Z}_{p}$. As $\mathbb{Z}_{p}$ is a domain, we know that the $\mathbb{...
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A badly convergent p-adic series from one of the Schikhof's excercises

W. H. Schikhof in his book on Ultrametric calculus suggests to solve the following problem: Exercise 23.J (van Hamme) Use the ideas of the previous excercise to show that in $\mathbb Q_p$ ($p \neq ...
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Do the number of degree p extensions of p-adic fields lie in a recursive sequence? And if so, why?

I noticed something on this page, that may just be coincidental: http://www.lmfdb.org/LocalNumberField/ From inspecting the table there, you can conclude that most of the interesting extensions of ...
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Completions of number fields

I would like to prove a statement about completions of number fields, but I'm running into a problem. The statement I want to prove is Let $L/K$ be a Galois extension of number fields, $p$ a prime ...
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What does extra zero of an $L$-function mean?

This is a very vague question. What does an extra zero of an $L$-function mean? There are lots of papers written on this topic, investigating the extra/exceptional zeros of various $p$-adic $L$-...
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Finite extensions of $\mathbb Q_p$ are exactly completions of numberfields

I read that every finite extension of $\mathbb{Q}_p$ is in fact a completion of a numberfield K with a place of K. I also heard that this is a consequence of Krasner´s Lemma. Do you have any hint how ...
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Why must a zero of $f \in \mathbb{Z}_p[X_1, \dots, X_m] ~ (\text{mod } p^n)$ be simple in order to lift to $\mathbb{Z}_p$?

In chapter II, section 2.2, of J-P. Serre's A Course in Arithmetic, we have the following theorem: Theorem 1: Let $f \in \mathbb{Z}_p[X_1, \dots, X_m]$, $x = (x_i) \in \left( \mathbb{Z}_p \right)^...
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How to show that some equation does not have any solutions in $\mathbb Q_p^2$.

Is it true that if an equation has solutions in $\mathbb Q^2$, it has a solution in $\mathbb Q_p^2$ for all primes $p$? For example, if $f(a, b) = a^2 - 2b^2$, the only solution of $f(a,b) = 0$ in $\...
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What constitutes a good reading course in $p$-adic number theory?

I have had a course in number theory where I studied Marcus and also a course in differential geometry. I have read Koblitz's introductory book on $p$-adic numbers. I am roughly interested in both $p$-...
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Find the first six terms in a reduced coherent sequence defining $i$ in $\mathbb{Z}_5$.

Find the first six terms in a reduced coherent sequence defining $i$ in $\mathbb{Z}_5$. I need to use the sequence $a_k=2^{5^{k-1}}$ but not sure how to? Any hints?
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Congruence subgroup of $\mathbb{GL}_n(\mathbb{Z}_p)$

In course of my research I met the following situation : 1) I have a bunch of open subgroup (so of finite index) in $\mathbb{GL}_{n}(\mathbb{Z}_p)$. 2) My groups arises naturally as stabilizers of ...
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$4$th root of unity: 5-adic

For $k \geq 1$, let $x_k = a^{p^{k-1}}$. Taking $p = 5$ and $a = 2$, find the first six terms in a reduced coherent sequence defining a $4$th root of unity (i.e. $\sqrt{−1}$) in $\mathbb{Z_5}$, and ...
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55 views

$\mathrm{ord}_p(x)$ and convergence in $\mathbb{Q}_p$

Let $x=\frac{22}{7} \in \mathbb{Q}$. (a) Find $\mathrm{ord}_p(x)$ for all primes $p$. $\mathrm{ord}_2(x)=1,\ \mathrm{ord}_{11}(x)=1,\ \mathrm{ord}_7(x)=-1$ and $\mathrm{ord}_p(x)=0$ for all ...
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$3$-adic expansion of $- \frac{9}{16}$

I get the $3$-adic expansion to be $1+1 \cdot 3+2 \cdot 3^2 +2 \cdot 3^3 + 0 \cdot 3^4+\cdots$. I'm trying to work out a pattern of the coefficients and think it is $1, 1, 2, 2, 0, 0, 1, 1, 2, 2, 0, 0,...
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Explanation of the 5-adic expansion of 2

Why is the $5$-adic expansion of $2 = 2 + 0\cdot5^2 + 0\cdot5^3 + ...$? I've done some working with powers of $5^k$ for $k=1,2,...$ and got that the 5-adic expansion is $2+2\cdot5+2\cdot5^2 + 2\cdot5^...
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$5$-adic expansion

Given $x = 2+1\cdot5+3\cdot5^2 +2\cdot5^3 +\ldots ∈ \mathbb{Z_5}$, find $\frac{1}{x}$, expressing it similarly as a $5$-adic expansion. (First 4 digits only). I'm new to p-adic numbers and was ...
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Kummer map and cohomology group for an elliptic curve

Let $E=E_q$ be the Tate ellipitc curve over a finite extension $K$ of $\mathbb{Q}_p$ for a $q$. Let $T$ be its p-adic Tate module. Let $\mathfrak m$ be the maximal ideal in $K$. I saw in this paper (...
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${\mathbb{Q}_p^*}^2$ is open in $\mathbb{Q}_p^*$

Show that the set of squares in $\mathbb{Q}_p^*$ is open in $\mathbb{Q}_p^*$. Here $\mathbb{Q}_p$ is the $p$-adic numbers and $\mathbb{Q}_p^*$ is the set of units in $\mathbb{Q}_p$. I know that $\...
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$5$-adic expansion of $−2$

Let $\alpha = \sum_{n=0}^{\infty} a_n\ p^n$ be the expansion of a p-adic unit. So $0<a_0<p$ and $0\leq a_n<p$ for $n \geq 1$. Show that $\beta = -\alpha$ has the expansion $\beta = \sum_{...
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If $p>3$ a prime number then $\binom {p-1}{\frac{p-1}{2}} \equiv (-1)^{\frac{p-1}{2}} 4^{p-1} \pmod {p^3}$

Here is one of Morley's theorem in number theory. My idea is to begin in $\mathbb{Z/pZ}$ : $\binom {p-1}{\frac{p-1}{2}} = \frac{(p-1)!}{(\frac{p-1}{2})!(\frac{p-1}{2})!}=\frac{(p-1)!}{(-1)^{\frac{p-...
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$\exp(2)$ does not converge $2$-adically.

We have $\exp(2)= \sum_{i=0}^n {\frac{2^n}{n!}}$. I am trying to show that $\exp(2)$ does not converge $2$-adically. i.e. I need to show $\nu_2 (\frac{2^n}{n!})$ does not tend to $\infty$ as $n\to \...
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Equality of $p$-adic fields

How can I prove that $\mathbb{Q}_3(\sqrt{2})=\mathbb{Q}_3(\sqrt{5})$. I only did modulo reduction, but how to prove it directly?
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Prove that a $p$-adic integer $x$ is divisible by $p^n$ if and only if $x_n = 0$

Let the ring of $p$-adic integers be the projective limit $$ \mathbb{Z}_p = \varprojlim_{n\in\mathbb{Z}_{\geq 1}}(\mathbb{Z}/p^n\mathbb{Z}), $$ and denote an element $x\in\mathbb{Z}_p$ as a sequence $...
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p-adic expansion of reciprocals

I can't seem to find an explanation on how to find the p-adic expansion of a reciprocal, for example, the 5-adic expansion of $ \frac 1 {10}$. Would anyone be able to give me a general method for ...
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Isomorphism of algebraic closure of p-adics with their completion

Consider the following fields: 1) $\mathbb{C}$ the complex numbers 2) $\overline{\mathbb{Q}}_p$ 3) $\mathbb{C}_p : = \hat{\overline{\mathbb{Q}}_p}$ They are all the same cardinality, algebraically ...
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Existence of inverse in $2$-adic ring

I need to prove: If $p\equiv 1 \bmod 16$ then there exists $x\in \mathbb{Z}_2$ ($2$-adic ring) so that $$px^4=1.$$ I'm not sure how to start this. I thought maybe to use some results on ...
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Solving matrix equations over $p$-adic rings

Let $F/\mathbb{Q}_p$ be a finite extension, and let $\mathcal{O}_F$ be its ring of integers. Now for $ 0< i,j \le r$ let $B_{i,j} \in \mathit{Mat}_{n \times n}(\mathcal{O}_F)$ be some matrices and ...
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Show $|x^2 + y^2 - 7|_7 \geq \frac{1}{7}$ in p-adics

Working in the $7$-adic numbers, I want to show that for any $x, y \in \mathbb{Q}$, we can guarantee that $|x^2 + y^2 - 7|_7 \geq c$ for some $c$. By some amount of trial and error, I've guessed that ...
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Euler totient function and unramified extension of $\mathbb{Q}_p$. A clarification.

I'm studying the construction of unramified extensions, and many references say that it's enough to attach the $p^n-1$ primitive root of unity to $\mathbb{Q}_p$ in order to obtain the unique degree $n$...
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What is the significance of Coleman maps arising in Iwasawa thoery?

I have come across two instances of "Coleman map" Let $E$ be an elliptic curve defined over $\mathbb{Q}_p$. Let $k_\infty$ be the unique $\mathbb{Z}_p$ extension of $\mathbb{Q}_p$ contained in $\...
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Connected Components of p-adic rationals

Notation: $p$ - a prime integer, $\Bbb{Z}_p$ - set of $p$-adic integers, $\Bbb{Q}_p$ - set of $p$-adic rationals, $\Bbb{Q}$ - set of rationals, $\Bbb{R}$ - set of reals. While reading up on $p$-...
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Is I-adic completion a ring epimorphism?

Let $R$ be a commutative ring and let $I \subset R$ be an ideal. For any $n \ge 1$, the ring homomorphism $R \rightarrow R/I^n$ is surjective, hence an epimorphism in the category of rings. What about ...
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How algebraically restrictive is the structure of $\Bbb Q_p$?

My question is a little difficult to explain, but I will try to state it. First, let me talk about the field $\Bbb R$ of real numbers. Let $K$ be the maximal real algebraic extension of $\Bbb Q$, that ...
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P-adic norm/valuation of n!

I am trying to figure out the valuation of n! with respect to some p-adic norm p. This is part of a proof about the convergence behavior of $x^n /n!$ in the p-adics. When I try and expand out n into ...
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examples of unramified extensions of $\mathbb{Q}_p$

For every local field $K$ and natural number $n$ coprime to $K$'s residue characteristic, there is a unique unramified extension $L/K$ of degree $n$. Let's take $K=\mathbb{Q}_p$. What are some ...
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A fundamental tool used in the study of Diophantine equations…

Notations: $K$ is a perfect field, $\overline K$ an algebraic closure and $V\subseteq \mathbb P^n(\overline K)$ is a projective variety on $\overline K$. If $V$ is defined over $K$, in symbols $V/K$,...
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Are the $p^n$-adic numbers isomorphic to the $p$-adic numbers?

In another recent question, the 10-adic numbers came up (along with the usual issues of not really being a field due to 10 not being prime, etc). I had a thought: ordinary binary numbers (either ...
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Why does an argument similiar to 0.999…=1 show 999…=-1?

I accept that two numbers can have the same supremum depending on how you generate a decimal representation. So $2.4999\ldots = 2.5$ etc. Can anyone point me to resources that would explain what the ...
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Completion of an extension $K|\mathbb{Q}$ w.r.t. a non archimedean abs. value, isomorphic to $K\cdot \mathbb{Q}_p$

As the title suggests I want to prove the following result: given $\mathfrak{p}|(p)$, prime ideal of $\mathcal{O}_K$ over $p$, we have the canonical absolute value induced by it on $K$, and we can ...
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$p$-adic digits via character sums

Let $p$ be a prime and let $n = \sum_{k=0}^\infty n_k p^k$ be a $p$-adic integer with each $0 \leq n_k \leq p-1$. Fix $0 \leq c \leq p-1$. Is there a way to check whether the $i$-th digit $n_i$ equals ...
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Will someone kindly explain Kato's dual exponential map?

I am reading this article by Rubin. Will somebody how to derive the formula given in equation 2 of section 5? It states thus: $z$ corresponds to the map $$x\mapsto Tr_{\mathbb{Q}_{n,p}/{\mathbb{...
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Proving that $d(a,b)=p^{-n}$ is a metric for $\mathbb{Q}$

I have the following task: If we have the metric $d:\mathbb{Q}\times\mathbb{Q}\rightarrow\mathbb{R}$, so that $d(a,a)=0$ and $d(a,b)=p^{-n}$ always when $a-b=p^nh/k$, where $h,k,n\in\mathbb{...
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ducci sequences using p-adic evaluation

considering a starting sequence of integer $(a_1,a_2,...,a_n) \in \mathbb{Z^n}$ let it apply the ducci operator $D$ that act like this \begin{equation} \mathbb{Z^n}\rightarrow \mathbb{Z^n} \\ (a_1,...
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basis of the p-adics $\mathbb{Q}_p$ as a $\mathbb{Q}$-vector space

The p-adics $\mathbb{Q}_p$ are uncountable (because they can be represented by infinite strings of integers in $[0,p-1]$) and hence must be infinite dimensional as a vector space over $\mathbb{Q}$. ...
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Is there a better representation than p-adics for exact computer arithmetic?

I stumbled across Quote notation and went hog wild. But when I stumbled on a technical detail I received a very discouraging comment: I think those authors may have been a bit short-sighted, ...
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Prove that if an integer $z$ is not divisible by $p$, then it is invertible in the $p$-adic integer ring $\mathbb{Z}_p$.

Let $p$ be a prime number. Define the $p$-adic valuation on $\mathbb{Z}$ as $v_p(p^kx) = p^{-k}$ where $x$ is not divisible by the prime $p$. Let $\mathbb{Z}_p$ (the $p$-adic integers) be a ...
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p-adics $\mathbb{Q}_p$ is a field if and only if $p$ is a power of a prime

I want to show that the ring $\mathbb{Q}_p$ is a field for any prime $p$, so I want to show that every nonzero element has an inverse. I thought of the following argument, but I can't seem to locate ...
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Quick question about number of positive summands in a sum of $p$-adic integers

I've started reading recently on $p$-adic numbers online. Forgive me if the question is silly. Let $\mathbb{Z}_p$ be the ring of $p$-adic integers and let $a_1, \ldots, a_k \in \mathbb{Z}_p$. If $...