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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Reference requested: 'decomposition' of Haar measure on the adeles.

Since the adeles $\mathbb{A}$ (with addition) are a locally compact Hausdorff topological group there exists a Haar measure $\mu$. Now people claim that it can be normalized such that for every ...
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44 views

A non-continuous p-adic representation

I am looking for an example of a non-continuous homomorphism $$G \to GL_r(\mathbb C_p)$$ from a profinite (topologically finitely generated) group $G$, where $\mathbb C_p$ is the completion of an ...
4
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1answer
411 views

Multiplicative Haar measure on $\mathbb{Q}_p$?

I have read in a book that if one takes $\mu$ to be the additive Haar measure on $\mathbb{Q}_p$, the p-adic rationals, then $$\nu(A) := \int_{A} 1/|x|_p dx$$ is a multiplicative Haar measure on ...
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Relationship between Connes trace formula and Weil's trace formula

Connes trace formula $$ \mathrm{Tr}\,{U(h)}=2h(1)\ln\Lambda + \sum_{v} \int d^{*}x \frac{h(u^{-1})}{|1-u|} $$ Weil's trace $$ \int_{C}h(u)|u|d^{*}u- ...
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284 views

Local solutions of a Diophantine equation

I am trying to prove that the equation $$3x^3 + 4y^3 +5z^3 \equiv 0 \pmod{p}$$ has a non-trivial solution for all primes $p$. I am sure that this is a standard exercise, and I have done the easy ...
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119 views

Unramified extension is normal if it has normal residue class extension

Let $K/F$ be an unramified extension such that $\rho_K / \rho_F$ (the corresponding extension of residue classes) is normal. Prove $K/F$ is normal. I guess I need to do some polynomial lifting, but ...
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2answers
472 views

Question about $p$-adic numbers and $p$-adic integers

I've been trying to understand what $p$-adic numbers and $p$-adic integers are today. Can you tell me if I have it right? Thanks. Let $p$ be a prime. Then we define the ring of $p$-adic integers to ...
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346 views

Are all $p$-adic number systems the same?

After just having learned about $p$-adic numbers I've now got another question which I can't figure out from the Wikipedia page. As far as I understand, the $p$-adic numbers are basically completing ...
3
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1answer
167 views

A property of non-Archimedean metrics

I have recently been reading about non-Archimedean metrics on fields (in Koblitz: $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions), and came across the exercise: Prove that a norm $\|.\|$ on ...
9
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134 views

diagonalizing a matrix over the $\ell$-adics

Let $M$ be a $2 \times 2$ matrix with coefficients in $\mathbb{Z}_{\ell}$ whose characteristical polynomial is $$ P(T) = T^2- (a+d) T + (ad-bc). $$ I've encountered the following assertion: If ...
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456 views

Why does the equation $x^2-82y^2=\pm2$ have solutions in every $\mathbb{Z}_p$ but not in $\mathbb{Z}$?

I have been working on an exercise in H. P. F. Swinnerton-Dyer's book, A Brief Guide to Algebraic Number Theory. The question is like this: Show that $x^2-82y^2=\pm2$ has solutions in every ...
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1answer
116 views

Lifting additive characters

Let $K$ a finite extension of $\mathbb{Q}_p$ ($p$ prime different from 2) and let $G_K$ the absolute Galois group of $K$. Let $\bar{u} : G_K \longrightarrow \mathbb{F}_p$ a continuous additive ...
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134 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 = ...
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2answers
503 views

The $p$-adic integers as a profinite group

How to prove that if $\mathbb{Z}_p$ is the set of $p$-adic integers then $\displaystyle{\mathbb{Z}_p=\varprojlim\mathbb{Z}/p^n\mathbb{Z}}$ where the limit denotes the inverse limit? $\mathbb{Z}_p$ is ...
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400 views

Unramified p-adic extension implies Galois

I am looking for a short proof that if $L \supset K$ are finite extensions of the p-adic numbers $\mathbb{Q}_p$, then if $L/K$ is unramified, $L/K$ is Galois. I think the proof is related to somehow ...
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1answer
113 views

Choosing an isomorphism $\tau:\overline{\mathbf{Q}}_p\simeq\mathbf{C}$; how do things depend on choice of $\tau$?

I sometimes see arguments that begin by choosing an isomorphism of fields $\tau:\overline{\mathbf{Q}}_p\simeq\mathbf{C}$, and then defining some property in terms of this isomorphism. I'm not so ...
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187 views

Line in a proof on p69 in Cassel's Local Fields

I'm trying to read the proof of LEMMA 6.1 (Nagell) Let $u_n$ be defined by $u_0=0$, $u_1=1$ and $u_n=u_{n-1}-2u_{n-2} \hspace{20pt} (n\geq 2)$. Then $u_n=\pm1$ only for $n=1,2,3, ...
7
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4answers
370 views

Divergent series and $p$-adics

If we naïvely apply the formula $$\sum_0^\infty a^i = {1\over 1-a}$$ when $a=2$, we get the silly-seeming claim that $1+2+4+\ldots = -1$. But in the 2-adic integers, this formula is correct. Surely ...
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318 views

Why is $O_K\otimes \mathbb{Z}_p\cong \oplus_{\mathfrak{p}|p}O_{K,\mathfrak{p}}$?

In my old number theory notebook this is stated as a fact. However, I ran into problems when I tried to prove it. First let me state the (supposed) theorem accurately: Theorem (?) Let $K$ be a ...
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306 views

What are the branches of the $p$-adic zeta function?

I'm reading the book $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions by Neal Koblitz. In it, Koblitz wants to iterpolate the Riemann Zeta function for the values $\zeta_p(1-k)$ with $k \in ...
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1answer
389 views

P-adic integers

Show that $\frac{2}{p-1}$ is a $p$-adic integer and find its p-adic expansion. P-adic numbers really make little sense to me so any help explaining what to do and why would be really appreciated. ...
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276 views

The polynomial $x^p - x -1/p$ over $\mathbb{Q}_{p}$

I know that the polynomial $f(x) = x^p -x - \frac{1}{p} \in \mathbb{Q}_{p}[x]$ is irreducible. So, let $\alpha$ be a root of $f(x)$, and $K = \mathbb{Q}_p(\alpha)$. Let $O_K$ be the valuation ring of ...
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538 views

Newton polygons

This question is primarily to clear up some confusion I have about Newton polygons. Consider the polynomial $x^4 + 5x^2 +25 \in \mathbb{Q}_{5}[x]$. I have to decide if this polynomial is irreducible ...
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189 views

The number of solutions of the equation $x^n+y^n=1$ over $\mathbb{Z}_p$ with $\mathbb{Z}_p=\{\alpha\in \mathbb{Q}_p:|\alpha|_p\le 1\}$.

The number of solutions of the equation $x^n+y^n=1$ over $\mathbb{Z}_p$ with $\mathbb{Z}_p=\{\alpha\in \mathbb{Q}_p:|\alpha|_p\le 1\}=\{\sum_{i=0}^\infty a_ip^i:0\le a_i \le p-1\}$. I was trying ...
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2answers
330 views

What does the Haar measure on $\hat{\mathbf{Z}}$ look like?

What does the Haar measure on $\hat{\mathbf{Z}} = \prod_p \mathbf{Z}_p$ look like? Does it bear any relation to the "upper density" of a subset $S\subset\mathbf{N}$ defined by $m(S) = ...
8
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2answers
785 views

Is the algebraic closure of a $p$-adic field complete

Let $K$ be a finite extension of $\mathbf{Q}_p$, i.e., a $p$-adic field. (Is this standard terminology?) Why is (or why isn't) an algebraic closure $\overline{K}$ complete? Maybe this holds more ...
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236 views

Applications of the p-adics

As I was preparing to spend a month studying p-adic analysis, I realized that I've never seen the theory of p-adic numbers applied in other branches of mathematics. I can certainly see that the field ...
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1answer
100 views

Why is the trace map from a finite extension of $\mathbb{Q}_p$ continuous?

Let $k$ be a finite extension of $\mathbb{Q}_{p}$. Why is $tr_{k/\mathbb{Q}_{p}}$ a continuous map from $k$ onto $\mathbb{Q}_{p}$?
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1answer
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$\mathbb Z_p$-rank $\leq r_1+r_2-1$ in Leopoldt's conjecture

I am trying to understand Leopoldt's conjecture as formulated in section 5.5 of Washington's "Introduction to Cyclotomic Fields". For an algebraic number field $K$ let $E$ denote the global units, ...
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1answer
666 views

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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equation over $\mathbb{Z}_3$

Consider the ring $\mathbb{Z}_3$ of 3-adic integers. Does there exist a positive integer $n$ and a solution to $(X_1^2 + X_2^2 + \cdots + X_{n - 1}^2)^2 = 2X_n^4$ in $\mathbb{Z}_3^n$? If so, what is ...
4
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1answer
184 views

Is there a $p$-adic version of Liouville theorem?

That is, if a function $f$ is analytic and bounded in all $K$, a $p$-adic field (or more generally a complete non-archimedean field), has to be constant? And does the theorem work for functions on ...
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191 views

Easy way to determine the primes for which $3$ is a cube in $\mathbb{Q}_p$?

This is a qual problem from Princeton's website and I'm wondering if there's an easy way to solve it: For which $p$ is $3$ a cube root in $\mathbb{Q}_p$? The case $p=3$ for which $X^3-3$ is not ...
4
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1answer
195 views

Equivalence of Euclidean norm used in Ostrowski's Theorem

In a proof of Ostrowski's theorem (The only nontrivial norms on $\mathbb{Q}$ are $\|\cdot\|_{p}$ and $\|\cdot\|_{\infty}$), we come to the point where we have shown a certain norm, $\|\cdot\|$, has ...
4
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1answer
505 views

Is $\mathbb Q_r$ algebraically isomorphic to $\mathbb Q_s$ while r and s denote different primes?

It is obvious that $\mathbb Q_r$ is topologically isomorphic to $\mathbb Q_s$ while $r$ and $s$ denote different primes. But I really don't know whether it is true in the aspect of algebra. As I ...
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1answer
117 views

Quadratic form over the dyadic numbers

I would like to know whether $q=\langle 3,3,11\rangle$ (a diagonal ternary form) represents $2$ over $\mathbb{Q}_2$ (i.e. whether there exist $x,y,z\in\mathbb{Q}_2^\times$ such that $q(x,y,z)=2$). I ...
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363 views

Isotropy over $p$-adic numbers

Over what $p$-adic fields $\mathbb{Q}_p$ is the form $\langle3, 7, -15\rangle$ isotropic?
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2answers
245 views

Hilbert Symbols when $K =$ the $p$-adic numbers

How can I show that the Hilbert Symbol is bimultiplicative, when the local field is the $p$-adic numbers? Everything I can find just sort of asserts bimultiplicativity without much proof, so I'm ...
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2answers
358 views

Square roots in the $p$-adics

Suppose I want to know whether $\sqrt{7}\in\mathbb{Q}_5$, or more generally, whether $\sqrt{n}\in\mathbb{Q}_p$ for $n\in\mathbb{Z}$, $p$ an odd prime. What are the techniques for determining this? Am ...
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1answer
539 views

Extending the p-adic valuation

Given a prime $p$, the $p$-adic valuation on the field $\mathbb{Q}$ is the map $\nu:\mathbb{Q}^*\to\mathbb{Z}$ given by $\nu(p^ka/b)=k$, where $a,b$ are prime to $p$. I want to consider extensions ...
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1answer
133 views

Discontinuous functions from $\mathbb Q_p$ to $\mathbb R$

The question is how we construct a function $f:\mathbb Q_p\to\mathbb R$ so that $f$ is discontinuous at every $x_0\in\mathbb Q_p$.
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1answer
166 views

Valuations on number fields

I'm trying to explicitly compute modular representations of some finite groups -- the easiest example to discuss is the cyclic group $C_3$ when $p=3$. The three ordinary irreducible modules for $C_3$, ...
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449 views

How many $p$-adic numbers are there?

Let $\mathbb Q_p$ be $p$-adic numbers field. I know that the cardinal of $\mathbb Z_p$ (interger $p$-adic numbers) is continuum, and every $p$-adic number $x$ can be in form $x=p^nx^\prime$, where ...
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1answer
97 views

Nice description of $(1+2\mathbb{Z}_2)^{2^k}$?

For $p\neq 2$ it's easy to prove through the log/exp-correspondence that $$(1+p\mathbb{Z}_p)^{p^k}=1+p^{k+1}\mathbb{Z}_p.$$ This gives an easy way to compute the groups ...
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146 views

Can this function extend to a measure?

I believe I have one final question on a function I've been thinking about. To set up, let $\mathbb{Q}_p$ be the $p$-adic numbers, and let $B(x,r)$ denote the closed balls $$ ...
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1answer
203 views

$p$-adic intermediate value theorem

Is there a $p$-adic analogue to the intermediate value theorem? I know there is a notion of convex sets in the $p$-adic context but can we hope for an intermediate value theorem in this context?
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204 views

Can I build a finitely additive function on $\mathbb{Q}_p$?

This is partially motivated by a question I saw earlier here, Does such a finitely additive function exist? I've been reading about the topology of $\mathbb{Q}_p$ in Knapp's Advanced Algebra in ...
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2answers
1k views

Why are closed balls in the $p$-adic topology compact?

I was skimming through some of this paper Measurable Dynamics and Simple $p$-adic Polynomials out of curiosity. A few pages in, the author claims that closed balls are both open and compact sets in ...
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2answers
207 views

Balls in ultrametric spaces

Consider a ball $B$ in the ultrametric space $X=K^n$ where $K$ is a $p$-adic field. (Recall that a ball in $X$ is a set of the form $\lbrace x \in K^n \mid \|x\|:= \max_{i=1}^n (\mid x_1 \mid, \dots, ...
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1answer
278 views

Create an order relation over the field of p-adic numbers

I've come to know that you can't define an order relation over the field of p-adic numbers that is compatible with the addition and multiplication according to the ordered field axioms. I was ...