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-adic expansion of rational

I want to find the 7adic expansion of 1/4. I found that this is …1515152, by using the algorithm of finding 1/4 = k + 7q for each digit. Is this correct?
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A question about the definition of $p$-adic pseudo-measure.

Let $\mathfrak B$ be a profinite abelian group and let $\Lambda(\mathfrak B)$ be defined as the inverse limit $\varprojlim \mathbb Z_p[\mathfrak B/ \mathcal H]$ where the inverse limit is taken with ...
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Changing uniformizer of $p$-adics

In the theory of $p$-adic fields typically a uniformizer $\pi$ is chosen that generates the maximal ideal, $m$. And a few theorems later it can be shown that every element $x \in O$ of the ring of ...
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Proving that $\mathbb{Q}_p$ is not formally real.

I've been looking for a concrete proof without results. The only hint that I have found says: 1) $\mathbb{Q}_2$ contains a square root of $-7$. 2) $\mathbb{Q}_p$ ($p>2$) contains a square root of ...
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How do I compute the first few digits of √11 in $\Bbb Q_5$ [closed]

Compute the first few digits of √11 in $\Bbb Q_5$
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II. p-adic equations [2.1. Solutions] (J.-P. Serre)

Currently, I'm reading a chapter about p-adic equations in A Course in Arithmetic by Jean-Pierre Serre and I have a hard time understanding it. My questions/thoughts are in textboxes like these. ...
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Limit of p-adic numbers

Let $\alpha\in\mathbb{Z}_p^\times$. I read somewhere that the limit of $\alpha^{n!}$ as $n\rightarrow\infty$ is equal to one. Can someone explain to me why this is? Thanks!
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Proving that $-1$ has no square root in $\mathbb Z_3$ (3-adic integers)

I want to prove that $-1$ has no square root in $\mathbb Z_3$ (3-adic integers). By the definition of inverse limit, we know that there is a ring homomorphism $\phi$ from $\mathbb Z_3$ to $\mathbb ...
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Rational sum of the $p$-adic series

Koblitz states (as one of the excercises to chapter 2) that whenever we are given an integer $k > 0$ and prime $p$, the series $$ f(p, k) = \sum_{n=0}^\infty n^kp^n $$ converges in $\mathbb Q_p$ ...
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Stronger form of Hensel's lemma?

Let $f \in \mathbb{Z}_p[x]$ and suppose $|f(a)|_p < |f'(a)|_p^2$ for some $a \in \mathbb{Z}_p$. Let $a_1 = a$, and for $n \ge 1$ let$$a_{n+1} = a_n - f(a_n)/f'(a_n).$$How do I see that this defines ...
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Subgroups of finite index of the split maximal torus of $GL_n(\mathbb{Z}_p)$.

Let $E_{i,j}$ be the $n \times n$ elementary matrices. Let $G=GL_n(\mathbb{Z}_p)$. Let $T_G$ be the split maximal torus of $GL_n(\mathbb{Z}_p)$. Let $\Theta$ be the subgroup of $T_G$ consisting of ...
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If $a \in \mathbb{Z}_5$ and $a \equiv \pm1 \text{ }(\text{mod }5)$, does there exist $x \in \mathbb{Z}_5$ where $x^2 = a$?

If $a \in \mathbb{Z}_5$ and $a \equiv \pm1 \text{ }(\text{mod }5)$, does there exist $x \in \mathbb{Z}_5$ where $x^2 = a$? I know we want to use Hensel's Lemma somehow to assess this question, but I'm ...
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Constructing Witt polynomials

I am reading about Witt vectors, and I keep seeing the following set of congruences often: For example, in these notes here, on page 3 we see the following congruences: \begin{align} ...
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P-adic expansions of Integers

Is there a way to prove that $a$ is an element of the Integers, if $p$ is prime, $a$ is in $\mathbb Z_p$ and its $p$-adic expansion is eventually periodic? I know how to do this for the ...
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Please express the first 3 7-adic digits of a root of $x^3-1$ in $\mathbb{Z}_7$ other than 1.

Please express the first 3 p-adic digits of a root of $x^3-1$ in $\mathbb{Z}_7$ other than 1. Does this just mean find the 7-adic expansion of 2 or 4? Wouldn't their expansions just be 2 and 4?
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$p$-adic field with infinite residue field. [duplicate]

I am reading J M Fontaine's book where on page 7 the following definition is made: A local field($K$) is a complete discrete valuation field whose reside field($k$) is a perfect field of ...
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Majoration of the $p$-adic valuation of a factorial.

Let $p$ be a prime number. In order to prove a result on $p$-adic interpolation of iterates, I need to show the following: Lemma. Let $m$ be an integer, one has: ...
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Show that $\mathrm{SO}_3(\mathbb{Q}_p) \simeq \mathrm{SL}_2(\mathbb{Q}_p) $

I read in a paper that $\mathrm{SO}_3(\mathbb{Q}_p) \simeq \mathrm{SL}_2(\mathbb{Q}_p) $ this is counterintuitive / surprising since $\mathrm{SO}_3(\mathbb{R}) \not \simeq \mathrm{SL}_2(\mathbb{R}) $ ...
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Can the zeroes of a multivariate $p$-adic polynomial be bounded?

Multivariate real polynomials, as opposed to multivariate complex polynomials, can have bounded zero sets, i.e. $x^2+y^2-1$ in $\mathbb{R}^2$. This fails in $\mathbb{C}^n$ because $\mathbb{C}$ is ...
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Find a matrix of order $p$ in a subgroup of $\operatorname{GL}_n(\mathbb Z_p)$

Let $p$ be a fixed prime and $\mathbb Z_p$ the ring of $p$-adic integers. Consider the subgroup $G_n\subseteq \operatorname{GL}_n(\mathbb Z_p)$ given by all matrices $(a_{ij})_{ij}$ such that $$ ...
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Serre mass formula under extension

Suppose $K_1$ is a local field and $K_2$ is a totally ramified finite Galois extension. Let $e$ be a positive integer with $[K_2:K_1]|e$. Consider the set of isomorphism classes of $K_1$ extensions of ...
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“Logarithmic derivative of a p-adic number”

Coleman, in "Division Values in Local Fields", (Inventiones math. 53, 91 - 116 (1979)), says that In his work on cyclotomic fields Kummer observed that various formal operations on power series ...
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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: ...
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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 ...
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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 ...
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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 ...
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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 ...
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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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$\mathrm{ord}_p(x)$ 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, ...
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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 + ...
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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 = ...
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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 $\mathbb{Z/pZ}$ : $\binom {p-1}{\frac{p-1}{2}} = ...
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$\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 ...