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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12
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198 views

p-adic Lie groups vs. algebraic groups over $\mathbb{Q}_p$

I am somewhat confused about the following two concepts and the relations between them- One concept is a Lie group $G$ over the $p$-adic field. This is defined in a similar fashion to a (real) Lie ...
7
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88 views

Ramification group fixing an unramified extension

For a Galois extension of local fields $L/K$ with Galois group $G$, define the ramification groups $G_i$ by $$G_i = \{ \sigma \in G : \nu_{L}(\sigma(x) - x) \geq i+1 \text{ } \forall x \in ...
6
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72 views

A slightly stranger Hensel's Lemma

I'm trying to understand the solution to this problem. It came up when doing some revision. It is essentially to show that the conclusion of Hensel's Lemma holds if we have take a valuation ring $R$ ...
6
votes
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156 views

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 ...
5
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75 views

Neukirch's motivation for $p$-adic numbers

I've started reading Neukirch's Algebraic Number Theory book and at the beginning of Chapter II he starts his motivation for the $p$-adic numbers as follows: "The idea originated from the observation ...
5
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47 views

A p-adic integral

Let $(K,||)$ be a finite extension of $\mathbb{Q}_p$ of degree $d$ such that the restriction of $||$ to $\mathbb{Q}_p$ is the usual p-adic absolute value. Endow $GL_n(K)$ with the unique Haar measure ...
5
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276 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 $\bigcap_k O_k$ ...
5
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172 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 ...
4
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59 views

Question about topological properties of $\Bbb{C}_p$

It is known that the structure of $p$-adic integers, $\Bbb{Z}_p$ is homeomorphic to the Cantor set, and $\Bbb{Q}_p$ is homeomorphic to the one-point deleted Cantor set (as I know, I don't certain it.) ...
4
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158 views

Why is the Euclidean metric called the prime at infinity?

I've been studying p-adic analysis recently and after a bit of searching on the web, I haven't found an answer as to why the Euclidean metric is referred to as the 'prime at infinity', and given the ...
4
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139 views

Liftings in unramified extensions of $\Bbb Z_p$

[Edit : I have changed the formulation of the question. Sorry for the trouble] Here is a stupid question, maybe trivial. Let $p$ be a prime number, $q = p^n$ where $n$ is an integer, $R = ...
4
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65 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)?
4
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408 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 ...
4
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430 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 ...
3
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23 views

Determinant 1 matrix does not change p-adic measures

Let $f:\mathbb Z^d \rightarrow \mathbb Z^d$ be a linear map having determinant 1. Is there an obvious way to see that if $U\subseteq \mathbb Z_p^d$ is a measurable set, then the p-adic measure of $U$ ...
3
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41 views

Classical geometry vs. p-adic geometry

I am interested in learning more about $p$-adic analysis and some of the geometric theories that are commonly used. However, it seems that many of the tools there are developed from analogous ...
3
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47 views

the different of finite extension of p-adic numbers

Let $F=\mathbb Q_p$ be the field of p-aidc numbers. Let $\xi_n$ be a $n$-th primitive root of unity. Now consider the finite extension of fields $K/F$ where $K=\mathbb Q_p(\xi_n)$. I want to find the ...
3
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26 views

Iwasawa module: $O[[G]]$ is noetherian where $G$ is a compact $p$ adic Lie group

Let $G$ be a compact $p$ adic Lie group. Let $E$ be a finite extension of $\Bbb{Q}_p$ with ring of integers $O$. Then how to show that $O[[G]]$ is noetherian. I was reading the article here and on ...
3
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41 views

Herbrand Quotient: Formula for $|K^*/(K^*)^n|$

NB: Please restrict answers to hints and not solutions. Problem: Use the theory of the Herbrand quotient $q(A)=H^{0}(A)/H^{1}(A)$ to show that, if $K$ is a finite extension of $\mathbb{Q}_p$, and ...
3
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64 views

For $p\ge3$ there is no extension of $\mathbb Q_p$ with Galois group $S_4$

I'm trying to show that if $p\ge3$ is prime, then There is no extension $K$ of the field of $p$-adic numbers $\mathbb Q_p$ with Galois group $S_4$. I know that $K$ must have a subextension ...
3
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61 views

Lucas' Theorem for $p$-adic integers?

Does Lucas' Theorem hold for $p$-adic integers? More specifically, does it specifically hold for the case that, given a $p$-adic integer: $$x = x_0 + x_1p + x_2p^2 + \cdots = \sum_{i=0}^\infty ...
3
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64 views

Does a generalization of the Teichmuller-character for non-prime arguments exist?

Rereading an older article on Fermat-quotients in which I'd applied some p-adic-rationale I find now, that my method for the representation of bases $b$ which allow high fermat-quotients ...
3
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25 views

properties of certain semigroup action on $\mathbb{Z}/p\mathbb{Z}$

Suppose we have a polynomial $f \in \mathbb{Z}/p\mathbb{Z}[x]$, $f(x) = x^2 - x$. We are interested in elements $n \in \mathbb{Z}/p\mathbb{Z}$ such that after repeated application of f they ...
3
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31 views

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 ...
3
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102 views

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 ...
3
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193 views

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 ...
3
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61 views

$ \mathcal O_K/(p) \cong \mathbb F_p[T]/(T^n)$?

Let $K$ be a totally ramified extension of $\mathbb Q_p$ of degree $n$. Then $$ \mathcal O_K/(p) \cong \mathbb F_p[T]/(T^n) .$$ What is this isomorphism?
3
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38 views

Borel lemma on rationality

I am looking for an enlightening proof of the following fact: Let $F(t)\in \mathbb{Z}[[t]]$ and suppose $S$ is a finite set of places on $\mathbb Q$ containing $\infty$. If for every $v\in S$ $F(t)$ ...
3
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48 views

computation of an $H^2$

Let $p$ be a prime number and $\mathbb{Q}_p$ the field of $p$-adic numbers. Let $G_p$ be the absolute Galois group of $\mathbb{Q}_p$ and fix an absolutely irreducible representation $$ \rho : G_p \to ...
3
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78 views

Maximal compact subgroup of $GL_n(\mathbb C_p)$

It is known that the general linear group $GL_n(\mathbb Q_p)$ over the $p$-adic numbers has $GL_n(\mathbb Z_p)$ as a maximal compact subgroup and every other maximal compact subgroup of $GL_n(\mathbb ...
3
votes
0answers
59 views

Spectrum theorem for p-adic matrix analysis

Recently, I met a problem related to p-adic matrices in my research, the key of the problem can be summarized in the following way: 1: whether there exist spectrum theorem for p-adic matrix.\ 2: ...
3
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106 views

p-adic liftings on SAGE

I asked a question the other day: Multidimensional Hensel lifting which @Hurkyl kindly and very elegantly answered. A follow-on from this is that I have tried to implement exactly the "algorithm" ...
2
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25 views

Mixed vs Equal Characteristic Local Rings

This is more of a vague, intuitive type of question, so perhaps there isn't anything too concrete anyone can offer. I am trying to get a sense of precisely why working with local rings of equal ...
2
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36 views

Inverse limit of the following system.

Let $\mathcal{O}_L = \mathbb{Z}[\sqrt{-5}]$ be the number ring of $L=\mathbb{Q}[\sqrt{-5}]$, and $\mathfrak{p} =(2, 1+\sqrt{-5})$ a prime (and maximal ideal) in $\mathcal{O}_L$. What is the inverse ...
2
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22 views

Why is $m \infty$ the conductor of $K = \mathbb{Q}(\zeta_m)/\mathbb{Q}$?

Wouldn't this be saying that for all $p$ dividing $m$, $1 + p^{\operatorname{ord}_p(m)} \mathbb{Z}_p$ is contained in the group of local norms $N_{\mathfrak p/p}(K_{\mathfrak p})$, where $\mathfrak p$ ...
2
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23 views

Show that $U_i \rightarrow U_{i+e}, x \mapsto x^p$ is an isomorphism

Show that $U_i \rightarrow U_{i+e}, x \mapsto x^p$ is an isomorphism? Let $K$ be a finite extension of $\mathbb{Q}_p$ with uniformizer $\pi$, prime $\mathfrak p$, and ramification index $e = ...
2
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23 views

What is the normal core of $GL_2(\mathbb{Z}_p)$ in $GL_2(\mathbb{Q}_p)$?

What is the normal core of $GL_2(\mathbb{Z}_p)$ in $GL_2(\mathbb{Q}_p)$? (the normal core of $H$ in $G$ is the largest subgroup of $H$ which is normal in $G$. it is the intersection of all ...
2
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25 views

The “$GL_2(\mathbb{Z}[1/p])$-part” of an element in $GL_2(\mathbb{Q}_p)$

Is there a group homomorphism $\varphi:GL_2(\mathbb{Q}_p)\rightarrow GL_2(\mathbb{Z}[1/p])$ such that for every $x\in GL_2(\mathbb{Q}_p)$ we have $x^{-1}\cdot\varphi(x)\in GL_2(\mathbb{Z}_p)$? My ...
2
votes
0answers
50 views

$\mathbb{Z}_p$ as a module over $\mathbb{Z}_{(p)}$

I denote by $\mathbb{Z}_{(p)}$ the localization at a prime p, and by $\mathbb{Z}_p$ the p-adic integers. Question: what is the structure of $\mathbb{Z}_p$ as $\mathbb{Z}_{(p)}$- module? For example ...
2
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43 views

Lattices in $\mathbb{Q}_p^n$ with the same stabilizer

Consider the action of $GL_n(\mathbb{Q}_p)$ on $\mathbb{Q}_p^n$, and let $T$ be the diagonal torus. Let $\Lambda$ and $\Lambda'$ be full-rank sublattices ($\mathbb{Z}_p$-submodules of rank $n$) such ...
2
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28 views

Counting tamely ramified Galois extensions of $\mathbb{Q}_p$ with a given Galois group.

For a homework exercise, I'm to determine for each $p$ the number of non-isomorphic tamely ramified Galois extensions $K/\mathbb{Q}_p$ such that $\operatorname{Gal}(K/\mathbb{Q}_p) \cong ...
2
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203 views

Every Cauchy sequence converges

SENTENCE: The p-adic numbers are complete with respect to the p-norm, ie every Cauchy sequence converges. PROOF: Let $(x_i)_{i \in \mathbb{N}}$ a Cauchy-sequence in $\mathbb{Q}_p$. We want to show ...
2
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18 views

no quotient of $\mathcal{O}_{\mathbb{C}_K}$ modulo a proper ideal containing $p$ is a perfect $\mathbb{F}_p$-algebra?

I am reading the Notes on $p$-adic Hodge theory of O. Brinon & B. Conrad . Can someone explains the following things to me? «... no quotient of $\mathcal{O}_{\mathbb{C}_K}$ modulo a proper ideal ...
2
votes
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92 views

Show that i is an element of the p-adic integers if and only if p congruent to 1 mod 4

This exercise was given in a graduate course on Local Class Field Theory. We want to prove that $i\in \mathbb{Z}_p$ (the $p-$adic integers) if and only if $p\equiv 1 \mod 4$. For $\Rightarrow$, we ...
2
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75 views

Torsion points on elliptic curves, $E^1$

Let $E$ be an elliptic curve over $\mathbb{Q}_p$ and let $E(\mathbb{Q}_p) \supset E^0(\mathbb{Q}_p) \supset E^1(\mathbb{Q}_p) \supset \cdots$ be its $p$-adic filtration, where $E^n(\mathbb{Q}_p) = \{P ...
2
votes
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33 views

$\mathbb{Z}_p[\text{PGL}_2(\mathbb{F}_{p^n})]$-modules

Let $p$ be prime. Let $\mathbb{Z}_p$ be the $p$-adic integers. I'm intersted in $\mathbb{Z}_p[\text{PGL}_2(\mathbb{F}_{p^n})]$-modules of low rank over $\mathbb{Z}_p$ (rank $2$ is already very ...
2
votes
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74 views

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 ...
2
votes
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56 views

presentation of the inertia group of $p$-adic fields

It is known (see here) that the absolute Galois group of a $p$-adic number field $K$ (ie a finite extension of $\mathbb Q_p$) is topologically finitely presented for any odd prime $p$. Is it also ...
2
votes
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78 views

$p$-adic Integration

I am struggling to compute the following integral \begin{equation} \int_{\mathbb{Z}_p}\exp_p(|x|_p)d\mu \end{equation} where \begin{equation} \exp_p(x)=\sum_{n=0}^{\infty}\frac{x^n}{n!} \text{ defined ...
2
votes
0answers
79 views

Structure of $\mathbb{Q}_p(\zeta_p)$

Let $p \ne 2$ be prime number and denote by $\zeta_p$ the p-th root of unity. It's well known that $K = \mathbb{Q}_p(\zeta_p)$ has $t=1 - \zeta$ as prime element (generator of the Ideal $P_K = \{ x\in ...