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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Upper Numbering of Ramification Groups of Absolute Galois Groups for Totally Ramified Extensions

Suppose $K'/K$ is a totally ramified extension of $p$-adic fields of degree $e.$ A paper (p.9, line 15) I am reading seems to use the following formula for the upper numbering on the absolute galois ...
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zeroes of homogeneous analytic $p$-adic functions

I am trying to understand Lemme 2.1 page 3 of this paper by Pilloni. What is says (I think) is that if you have, for a a positive real number $w$, an analytic function $$ f : ...
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$p$-adic Fourier transforms and orthogonality relations

In $\mathbb{C}$, we have the following orthogonality relation $$ \int_{0}^{1} e^{2\pi i (m-n)x} dx = \begin{cases} 1 & \mbox{ if } m = n;\\ 0 & \mbox{ otherwise.} \end{cases} $$ Do we have ...
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Solutions in $\mathbb Q_p$ leads to solution for congruences equations?

Let $p$ be a prime number such that $p\equiv1\pmod 3$. Let $n$ be an integer such that the equation $x^3=n$ has a solution in $\mathbb Q_p$. In fact with our assumptions, the others solution are in ...
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Deducing Skolem-Mahler-Lech theorem from a $p$-adic interpolation result.

Let $p$ be a prime number, let $d$ be an integer greater than $1$ and let $f\in\mathbb{Z}_p[X_1,\cdots,X_d]^d$. I have already proved the following $p$-adic interpolation theorem: Theorem 1. ...
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Axiomatic Approach to p-adic rationals

There are several ways to construct the real numbers: completion of the rationals w.r.t. the euclidean metric, dedekind completion of the rationals, infinite continued fractions, etc. Each of these ...
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A number $a$ is a square in $\mathbf{Q}$ if and only if it is a square in $\mathbf{R}$ and $\mathbf{Q}_p$ for all primes $p$

Problem from Schikhof's Ultrametric Calculus. As I understand it, the intersection of $\mathbf{R}$ and all $\mathbf{Q}_p$ is just $\mathbf{Q},$ so it seems that $x^2-a$ having a zero in $\mathbf{Q}$ ...
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Limit of a sequence in $\mathbb{Z}_p$ (J.-P. Serre, p-adic equations)

In a proof of a theorem in chapter 2 "p-adic equations" in "A Course in Arithmetic" from J-P Serre there is one conclusion that I don't understand. Here is the theorem I'm talking about (excluding the ...
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Galois group action on etale cohomology groups

Let $X$ be a smooth and proper scheme over $Spec(\mathbb{Z}_p)$. Let $l$ be a prime number coprime to $p$. Then the proper base change theorem gives me an isomorphism ...
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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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Characters with values on the $p$-adic complex field $\mathbb{C}_p$?

Characters $\psi : G \to \mathbb{C}$ from abelian groups $G$ to the complex field $\mathbb{C}$ are well-known and appear all over. Is there an analogue for the $p$-adic complex numbers $\mathbb{C}_p$, ...
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Embedding a number field in $\mathbb{Q}_p$.

Let $K/\mathbb{Q}$ be a finitely generated field extension and let $(x_1,\cdots,x_m)$ be a transcendance basis of $K/\mathbb{Q}$. Using primitive element theorem, there exists $y\in K$ algebraic over ...
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A degree comparison problem over p-adic fields

Consider multivariate polynomial $p(x)\in\Bbb Q_p(x)$ and multivariate rational function $r(x)\in\Bbb Q_p(x)$ where $x=(x_1,\dots,x_n)$. Assume that degree in each variable is $1$ but total degree ...
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An example of a discontinuous “$\ell$-adic Galois representation”

Let $\mathbb{F}_p$ be a finite filed with $p$ elements, and $G=\mathop{\mathrm{Gal}(\mathbb{F}_p^s/\mathbb{F}_p)}$ be its absolute Galois group. $G$ is a pro-finite group, with the Krull topology, see ...
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Isometries of p-adic vector spaces?

In Euclidean space $\mathbb{R}^d$ there are ${d+1 \choose 2}$ independent isometries (translations and rotations). In other words the dimension of the Euclidean isometry group is ${d+1 \choose 2}$. ...
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Understanding a proof of a corollary in chapter 2 about invertibility of a p-adic integer (Jean-Pierre Serre)

In a proof of a corollary in chapter 2, there is a step I don't understand. Corollary 2: Suppose $p \neq 2$. Let $f(X) = \sum_j a_{ij}X_iX_j$ with $a_{ij} = a_{ji}$ be a quadratic form with ...
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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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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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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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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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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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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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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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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 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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$p$-adic valuation of harmonic numbers

For an integer $m$ let $\nu_p(m)$ be its $p$-valuation i.e. the greatest non-negative integer such that $p^{\nu_p(m)}$ divides $m$. Let now $H_n=1+\dfrac{1}{2}+ \cdots+ \dfrac{1}{n}$. If ...
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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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What is a “branch” of p-adic exponentiation?

I am reading p-adic Numbers, p-adic Analysis, and Zeta-Functions by Neal Koblitz. Please look at page 27. The definition of $f(x)=n^x$ is unambiguously defined when $n$ is $1 \mod p$. Next, for any ...
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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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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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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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What are the quadratic extensions of $\mathbb{Q}_2$?

How do you classify the non-squares in $\mathbb{Q}_2$? I've tried writing down expansions for "odd" numbers in $\mathbb{Z}_2$, but unlike in $\mathbb{Z}_p$, the n$^{th}$ term in the expansion is not ...
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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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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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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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Why does taking completions make number fields simpler?

I'm currently taking a course on Local Fields, and the local-theoretic picture seems to be significantly simpler than that of number fields. For example, If $K$ is a finite extension of $\mathbb ...
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What is the index of the $p$-th power of $\mathbb Q_p^\times$ in $\mathbb Q_p^\times$

In this book it is listed as an exercise to compute the index $[\mathbb Q_p^\times:(\mathbb Q_p^\times)^p]$. This exercise is appended to a section concerning the structure of unit-group filters, ...
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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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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 ...