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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A property of some sequences of natural numbers (and their binary representation)

Let's consider a sequence of natural numbers $a_n$, represented in binary, with the following properties: $\forall n \in \mathbb{N}$ the number $a_n$ is represented with $n$ binary digits $\forall ...
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135 views

Algebraic topology concepts in the $p$-adic setting

Can we expect that we might apply notions similar to "simple connectedness" and "multiple connectedness" to the $p$-adic setting, in spite of the fact that the standard topology of $\mathbb{Q}_p^n$ is ...
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1answer
298 views

Is there a $p$-adic version of the Riemann hypothesis?

Is there a $p$-adic version of the Riemann hypothesis or this does not make any sense?
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262 views

Intersection of nested open sets

Consider a nested sequence $O_1$, $O_2$, $\dots , O_k, \dots $ of open sets in $\mathbb{Q}_p^n$ such that $O_i \setminus O_{i+1}$ has empty interior. Under which conditions do we have $\bigcap_k O_k$ ...
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60 views

Green’s formula in p-adic integration

Is there an analogue of Green's formula in p-adic integration (with respect to the Haar measure)?
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1answer
80 views

normal groups of a infinite product of groups

I have a question regarding the quotient of a infinite product of groups. Suppose $(G_{i})_{i \in I}$ are abelian groups with $|I|$ infinite and each $G_i$ has a normal subgroup $N_i$. Is it true in ...
5
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3answers
252 views

Finding $p^\textrm{th}$ roots in $\mathbb{Q}_p$?

So assume we are given some $a\in\mathbb{Z}_p^\times$ and we want to figure out if $X^p-a$ has a root in $\mathbb{Q}_p$. We know that such a root must be unique, because given two such roots ...
4
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2answers
716 views

Roots of unity in a local field

The multiplicative group of a local field $K$ with valuation ring $\mathcal{O}$ and residue class field of $\overline{K}$ of degree $q=p^f$ splits as $K=\langle \pi\rangle\times \mu_{q-1}\times ...
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1answer
196 views

Integers in $p$-adic field

Let $K$ be a finite extension of $\mathbb Q_p$. How to prove that if an element of $K$ has non negative valuation then it is algebraic over $\mathbb Z_p$? I would like also a reference for this proof ...
6
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1answer
310 views

Integral closure of p-adic integers in maximal unramified extension

Let $\mathbb Q_p$ be the field of p-adic numbers, and let $\mathbb Q_p^{\text{unr}}$ be maximal unramified extension in some algebraic closure of $\mathbb Q_p$. My understanding is that $\mathbb ...
4
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1answer
130 views

Decomposition of $\mathbb{C}_p^*$

I'm looking for a topological group decomposition of $\mathbb{C}_p^*$. I know that I can write $\mathbb{C}_p^*\cong p^\mathbb{Q}\times \mathcal{O}_{\mathbb{C}_p}^* \cong p^\mathbb{Q}\times ...
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137 views

Showing that $|1-x^{p^n}|_p \leq p^{-p^n}$ when $|1-x|_p < 1$

Let $p$ be a rational prime, and let $x$ be an element of $\mathbb{Q}_p$ with $|1-x|_p < 1$. I want to show that $|1-x^{p^n}|_p \leq p^{-p^n}$ for all positive integers $n$, but I'm having a hard ...
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2answers
250 views

Representations of p-adic integers as certain infinite sums

One way to define the p-adic integers is as the $p$-adic completion of $\mathbb{Z}$. With some additional work, it can be shown that this is isomorphic to $\mathbb{Z}[[x]]/(x-p)$. Now, I know that ...
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1answer
113 views

Characterizing $\mathbb{Q}_p$ as an $I$-adic completion of $\mathbb{Q}$

It's known that the ring of p-adic integers $\mathbb{Z}_p$ can be characterized as the I-adic completion of $\mathbb{Z}$ for $I=(p)$. Is there any similar characterization for $\mathbb{Q}_p$ (i.e. an ...
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697 views

The p-adic numbers as an ordered group

So I understand that there is no order on the field of p-adic numbers $\mathbb{Q_p}$ that makes it into an ordered field (i.e.) compatible with both addition and multiplication. Now, from the ...
3
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1answer
239 views

Weird quotient of $\langle\mathbb Q,+\rangle$?

after looking at this question I came to think on one particular case. I'm wondering if maybe I've missed something on the way. If anyone could give it a look that would be great: We start by ...
7
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332 views

Roots of unity in $\mathbb{Q} _{11}$

Here $\mathbb{Q} _{11}$ denotes the 11-adic field. How can I show that the only root of unity of order 7 in this field is 1? Is it true that for any two distinct primes $p,q$, the only root of unity ...
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143 views

Characteristic polynomial and $p$-adic valuation

Suppose I had a linear operator $L$ whose characteristic polynomial was $f(x) = x^{n} + a_{1}x^{n-1} + \cdots + a_{n-1}x + a_{n}$. Furthermore, I also know that the eigenvalues of $L$ have $p$-adic ...
3
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1answer
165 views

The $p$-adic expansion of a function of $p$

Let $p\neq 2$ be prime. I am asked in a revision question to find the $p$-adic expansion of $(1+2p)/(p-p^3)$. The best I could do was find the $p$-adic norm, which I got as $p$ (please correct me if ...
3
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1answer
86 views

$p$-adic closures of infinite sets

Let $S\subsetneq\mathbb{Z}$ be an infinite set. Does there always exist a prime $p$ such that the closure of $S$ in the $p$-adic integers, $\mathbb{Z}_p$, contains a rational integer $n\notin S$? ...
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115 views

Are there any non-trivial rational integers in the $p$-adic closure of $\{1,q,q^2,q^3,…\}$?

If $p$ is prime and not a divisor of $q$, are there any non-trivial rational integers in the $p$-adic closure of the set or powers of $q$? Edit: $q$ is also a (rational) integer, not a $p$-adic.
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50 views

Uniqueness of extension of a norm on a ring to its completion

This is with reference to theorem 2.18 that appears here http://www.maths.gla.ac.uk/~ajb/dvi-ps/padicnotes.pdf Essentially the author says that if $N$ is a norm on a ring $R$ and $\hat{R}$ is the ...
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322 views

Ultrametric inequality

I am having trouble seeing the following consequence of the ultrametric inequality, which is supposed to be immediate. If $|x+y|\leq \max{\{|x|,|y|\}}$, then, equality holds when $|x|\neq |y|$. I ...
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85 views

Proving independence of log-function

Let $ \alpha_1, \ldots, \alpha_r $ be elements of a field $K \supseteq \mathbb Q$, which have the property that for $n_i \in \mathbb Z$, $i=1,\ldots,r$, it follows from $\alpha_1^{n_1}\cdots ...
4
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1answer
117 views

Subgroups of index 3 in $1+p\mathbb{Z}_p$

Let $p$ be a prime. I'm trying to compute the subgroups of index $3$ in $\mathbb{Q}_p^\times$ to enumerate some cyclic extensions using CFT. I've essentially reduced the problem down to finding the ...
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2answers
1k views

What do the $p$-adic roots of unity look like?

I know that $\mathbb{Z}_p$ has all the $p-1^{st}$ roots of unity (and only those). Is it true that mod $p$ they are all different? Meaning, is the natural map $\mathbb{Z}_p \rightarrow \mathbb{F}_p$, ...
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Potential computational questions that could be asked about p-adic numbers and Galois Theory

I have an exam on P-Adic integers (and a bit on Galois Theory) that my professor said would be very computational, but he never does any examples of the theorems he proves in class. He said the exam ...
4
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381 views

p-adic numbers and binomial coefficients

Let $\alpha\in \mathbb{Z}_p$ be an $p$-adic integer and define for $n\in \mathbb{Z}_{\geq 0}$ $${\alpha\choose n} := \frac{\alpha(\alpha-1)\cdot\ldots\cdot(\alpha-n+1)}{n!}.$$ This is again a ...
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163 views

Is there a notion of *p-adic Dedekind Domains*?

As we all know, the ring $Z_p$ can be constructed as the projective limit of the rings $Z/p^{n}Z$. Now is there any generalization such as the p-adic completions of a Dedekind Domain? This might be ...
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152 views

Does $\mathbb{Q}_p \cap \overline{\mathbb{Q}}$ depend on the embedding of $\overline{\mathbb{Q}}$ into $\overline{\mathbb{Q}_p}$?

I think the title pretty much says it all. I'm getting confused in the subtle parts of a proof, and would appreciate some help.
8
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606 views

Are all nonarchimedean valuations discrete?

I am studying valuation theory on the way to local class field theory, and the texts I have looked at immediately focus on discrete valuations in developing the theory of nonarchimedean valuations. ...
7
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2answers
659 views

$n$th powers in the p-adics

Suppose $K$ is a $p$-adic field (finite extension of the $p$-adics), and let $n$ be any integer (independent of what $p$ is). Define $U$ to be the set of all $x$ in $K$ such that $|x| = 1$ and such ...
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How far are the $p$-adic numbers from being algebraically closed?

A few days ago I was recalling some facts about the $p$-adic numbers, for example the fact that the $p$-adic metric is an ultrametric implies very strongly that there is no order on $\mathbb{Q}_p$, as ...
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2answers
423 views

Computing Newton polygons

I am looking at the first two examples in Paul Garretts notes on p-adic number theory. The first example is computing the Newton polygon of $x^5+2x^2+5$ over $\mathbb{Q}_2$. I think this is the lower ...
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688 views

Binomial coefficients: how to prove an inequality on the $p$-adic valuation?

In section 4 of the article by Afred van der Poorten's A Proof That Euler Missed ... the following inequality is used: $$\nu_{p}\displaystyle\binom{n}{m}\leq\left\lfloor\dfrac{\ln n}{\ln ...