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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112 views

Show that $y^2=x^3+1$ has infinitely many solutions over $\mathbb Z_p$.

I first compared it with how I would solve this over the real numbers. You would say: $y^2=\alpha$ has a solution for all $\alpha>0$, of which there are infinitely many. $x^3+1>0$ for all ...
4
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1answer
70 views

$p$-adic logarithm, $|\log_p(1 + x)|_p = |x|_p$?

Define the $p$-adic logarithm$$\log_p(1 + x) = \sum_{i =1}^\infty (-1)^{i-1}x^i/i.$$How do I see that if $p > 2$ and $|x|_p < 1$, then $|\log_p(1 + x)|_p = |x|_p$?
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25 views

Algebraic Closure and $p$-adic completion: do they commute? [duplicate]

I know that the algebraic closure of $\mathbb Q_p$, which I'll denote $\overline{\mathbb Q_p}$, is not metrically complete: there are $p$-adic Cauchy sequences that do not converge. (The example I ...
1
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1answer
26 views

Definition for non-degenerate module

[QUESTION] If $R$ be a ring, what is the meaning of a non-degenerate $R$-module? In a previous question post at (What is a non-degenerate module?), some experts said that if $M$ is a $R$-module such ...
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3answers
65 views

1/p in p-adic number system?

to expand $1/p$, I tried first letting $1/p = a+b*p+c*p^2+d*p^3+...$ and it is $1=a*p+b*p^2+...$ but I guess there's no way to make the equality hold. it's somewhat similar to dividing by 0. is it ...
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1answer
33 views

Property of valuation in $\Bbb{Z}_p$

Let $v$ denote the $p$-adic valuation in $\Bbb{Z}_p$. Let $a_1,a_2,a_3$ be three elements of $\Bbb{Z}_p$. Then I want to show that $$\min \{ v(a_1), v(a_2), v(a_1+a_2+a_3)\}=\min ...
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1answer
45 views

valuation of a particular element in $\Bbb{Z}_p$

Consider $x \in \Bbb{Z}_p$. Then I want to find the valuation of $(1+p)^x-1$. I think that $val_p((1+p)^x-1)=1+val_p(x)$. Is this right? Actually I want to prove that $min\{val_p(1+p)^x-1, ...
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2answers
35 views

About 2-adic representation of integers

How would I express -3 in 2-adic representation? Is it just revercimal calculation of binary expression of -3? like: -3 = -11 in binary, so using revercimal, -11. in binary = 01. ?
2
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1answer
36 views

Compute Limit of p-adic Cauchy Sequence

This has really been irking at me and I really should be able to do this but for some reason I can't so I'll ask on here. It is easy to compute the rational "equivalent" of a Cauchy sequence of the ...
6
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1answer
108 views

Projective limit involving p-adic numbers

Let $p$ and $q$ be distinct primes. What is the projective limit $$\varprojlim \mathbb R^2 / (p^n \mathbb Z \times q^n \mathbb Z)?$$ That's an exercise from Robert's book A Course in p-adic ...
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1answer
28 views

Do the adeles $\mathbb{A}$ have the same cardinality as the real numbers $\mathbb{R}$?

My question today is about the Adeles and whether they have the same cardinality as the real numbers. Certainly $\mathbb{R}$ has more elements than $\mathbb{Z}$. This is by Cantor diagonalization. ...
2
votes
1answer
41 views

$p$-adic sine series

This is a quick question about the domain of convergence of $p$-adic sine series. We define the $p$-adic sine function by the following power series $\sin_p(X) = \sum\limits_{n=0}^\infty ...
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0answers
28 views

Prove that the canonical p-adic expansion of $a\in\mathbb{Q}_p$ terminates

I am having trouble getting started with the following problem: Prove that the canonical p-adic expansion of $a\in\mathbb{Q}_p$ terminates (so $a_i = 0$ for all $i \geq N$) if and only if a is a ...
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1answer
47 views

7-adic expansion of a rational number

I know that every rational number has a unique 7-adic expansion, now I need help proving that $7/36=\sum\limits_{i=0}^nn7^n$ as a 7-adic integer. I tried using properties of this fraction, like adding ...
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1answer
35 views

How to calculate $\log_p x$ in p-adic analysis?

I'm studying p-adic analysis and recently I've learned about the p-adic logarithm function but I can't understand very well how the process of calculating the value should be done. As an exercise I'm ...
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1answer
48 views

Fast way to check if finite extension is unramified?

Consider the $5$-adic quadratic extension $\mathbb{Q}_{5}(\sqrt{5})/\mathbb{Q}_{5}$. I want to check if this extension is unramified, where unramified means that the corresponding extension of ...
4
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1answer
37 views

Solution of a equation in $\Bbb{Z}_p$

Let $m \in \Bbb{Z}_p$ be fixed. Let $a_1,...a_l$ be fixed integers. I am trying to find out solutions of the equation $m=x_1^{a_1}...x_l^{a_l}$ where $x_1,...,x_l\in \Bbb{Z}_p$. Here $x_1,...x_l$ are ...
4
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1answer
44 views

Quadratic extensions of p-adic rationals

In Alain Robert's A Course in p-adic Analysis, the author uses Hensel's Lemma to analyze quadratic extensions of $\mathbb{Q}_p$. He wants to calculate the index of $(\mathbb{Q}^*_p)^2$ in ...
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0answers
20 views

Prove that the sequence $(p^n)^{∞}_{n=1}$ for p a prime is a null sequence with respect to $| · |_p$.

Prove that the sequence $(p^n)^{∞}_{n=1}$ for p a prime is a null sequence with respect to $| · |_p$. Here is what I have: a null sequence is a sequence that maps to zero. $| · |_p=p^{-vp(\cdot)}$ ...
4
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1answer
109 views

A theorem on p-adic power series

This is a question about a proof of a theorem in many books on $p$-adic numbers and I don't seem to understand one of the directions. The theorem is Let $f(X) = \sum\limits_{n=0}^\infty a_nX^n \in ...
5
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0answers
79 views

Relation between Bombieri theorem and p-adic squares

Koblitz states in his book on p-adic numbers on page 84: Suppose that $\alpha \in \mathbb Q$ is such that $1 + \alpha$ is the square of a nonzero rational number $a/b$. Let $S$ be the set of all ...
3
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1answer
69 views

Tautological line bundle over rational projective space

Is the tautological line bundle over $\mathbb{Q}P^{n}$ a non trivial bundle? Here, $\mathbb{Q}P^{n}$ has the natural topology induced from the standard topology of $\mathbb{Q}$ as a subset of ...
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33 views

Square of a p-adic number is an integer.

This is a question for an assignment, so just some pointers in the right direction would be great. Suppose that $n \in \mathbb Z$ and that $\alpha \in \mathbb Q_p$ satisfies $\alpha^2 = n$. Prove ...
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2answers
65 views

Understanding proof that $\mathbb{Q}_{p}(\zeta)/\mathbb{Q}_{p}$ is unramified for $(n,p)=1$.

Problem Consider the extension $\mathbb{Q}_{p}(\zeta)/\mathbb{Q}_{p}$, where $\zeta$ is a $n$-th primitive root of unity and $(n,p)=1$. I want to show that $\mathbb{Q}_{p}(\zeta)/\mathbb{Q}_{p}$ is ...
2
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2answers
48 views

n-th power residues in p-adic fields

How to see that number of n-th power residue classes of a p-adic field K (elements of $K^*/K^{*n}$) is finite? I know how to prove using Hensel lemma that all p-adic integers sufficiently close to 1 ...
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1answer
24 views

Prove that $\mathbb{Z}_p = {a\in\mathbb{Q}_p:|a|_p\leq 1}$.

Let $\mathbb{Z}_p= {a\in\mathbb{Q}_p : a=\sum_{i=0}^{\infty}d_ip^{i},0\leq d_i\leq p-1\;\forall\;i\geq 1}$ be a subset of $\mathbb{Q}_p.$ In other words, $\mathbb{Z}_p$ is the set of all p-adic ...
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2answers
55 views

Minimal polynomial has coefficients in the valuation ring?

Question Let $K$ be a complete nonarchimedean valued field with valuation ring $\mathcal{O}$ and let $L/K$ be a finite extension. Let $\alpha$ be an element of $L$ and $f\in K[x]$ its minimal ...
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1answer
35 views

Cuspidal and Supercuspidal representation

Let $G$ be an algebraic group over a field $F$, and let $(\pi,V)$ be a smooth $G$-representation over an algebraic closed field $k$. Then $\pi$ is called a CUSPIDAL representation if $r(V)=0$ for any ...
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0answers
29 views

Let $p_1,p_2\in\mathbb{Z}$ be distinct prime numbers. Show that $|\cdot|_{p_1}\not\sim|\cdot|_{p_2}$

Let $p_1,p_2\in\mathbb{Z}$ be distinct prime numbers. Show that $|\cdot|_{p_1}\not\sim|\cdot|_{p_2}$ by finding a sequence in $\mathbb{Q}$ that is Cauchy with respect to $|\cdot|_{p_1}$ but not Cauchy ...
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0answers
21 views

Prove that the sequence $(s_n)$ in $\mathbb{Q}$,

Prove that the sequence $(s_n)$ in $\mathbb{Q}$, where $$\sum_{i=-m}^{n}d_i,p^i$$ with $0\leq d_i\leq p-1$ and $m\geq0$ represents a p-adic number. Here is what I have so far: Set $\epsilon>0$ ...
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1answer
24 views

Showing the sequence of partial sums of a power series is Cauchy under the p-adic norm

I want to show that the sequence $e_n = \sum_{i = 0} \frac{x^i}{i!}$ is Cauchy with respect to the p-adic norm when $p > 2$ and $|x|_p < 1$. So far, I have the estimate $|e_n - e_m|_p = |\sum_{i ...
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0answers
41 views

'Smooth' p-adic analysis (perhaps via toposes)

There are sensible theories of analytic functions on non-Archimedean fields (rigid analytic spaces, Berkovich spaces), but these are modeled after complex analysis. I'm curious to what extent there ...
2
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2answers
27 views

Prove that $a \equiv b \mod{p^j} \iff |a-b|_{p}\leq p^{-j}$

Prove that $a \equiv b \mod{p^j} \iff |a-b|_{p}\leq p^{-j}$ Typically, when dealing with a congruence I go to the division statement. i.e $$a\equiv b\mod{p^j}\Rightarrow p^j|a-b \;\;\;(\star)$$ ...
0
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1answer
32 views

Reconstructing formal groups from the p-map, realizing p-maps from formal group

Suppose $F$ is a formal group over $\mathbb{Z}_p$. There are few trivial condition that $f \in \mathbb{Z}_p[[t]]$, the power series representing the $p$-map should satisfy: 1)$f \equiv g(t^p) mod p$ ...
4
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2answers
86 views

Integral closure of the p-adic integers in a finite extension of the p-adic numbers

In Cassels' article "Global Fields", he uses the term "ring of integers" and the notation $\mathcal{O}_K$, where $K$ is a field with a non-archimedean valuation, to denote the ring of elements $x \in ...
4
votes
1answer
89 views

Quotient of $\mathbb{Z}_p$ by the rational integers

Let $p$ be a prime and let $\mathbb{Z}_p$ denote the $p$-adic integers. One has a canonical inclusion of rings $$\mathbb{Z}_{(p)}\longrightarrow\mathbb{Z}_p$$ given by identifying the rational ...
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1answer
31 views

Showing that $ord_p((p^n)!) = 1 + p + \dots + p^{n-1}$

I need to prove that $ord_p((p^n)!) = 1 + p + \dots + p^{n-1}$ ($ord_p$ is the $p-$adic valuation), which is essentially a nasty combinatorics problem. I want to count how many integers from $1$ to ...
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1answer
50 views

p-adic density of zeroes of a polynomial

I saw the following definition of the p-adic densities of zeroes of a system of polynomial equations: Definition: Suppose that we have a system of homogeneous polynomials of degree $d$, $ ...
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0answers
53 views

Valuations above $\mathbb{Q}(\alpha)$

Let $K$ be a field, we say that a function $\nu:K\to \mathbb{Z}_{\geq 0} \cup \lbrace \infty \rbrace$ is a valuation above $K$ if then followings hold for each choose of $x,y \in K$: 1 $\nu(x) = ...
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0answers
47 views

Closed subgroups of $Z_{p}^{\times}$

I was able to prove that any closed subgroup of additive group of $Z_{p}$ is of the form $p^{n}Z_{p}$ for some $n$. I asked the same question for the multiplicative group of units in $Z_{p}$, that is ...
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0answers
95 views

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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1answer
83 views

Examples of Cauchy sequences in the rational numbers that do not converge in said set with respect to the p-adic topology

I am working through a set of notes on Algebraic Number theory, and I find myself attempting to construct some examples of Cauchy sequences not converging in $\mathbb{Q}$, with respect to the p-adic ...
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1answer
22 views

Density of the image of the set $\lbrace (x,x), x\in \mathbb{Z} \rbrace $ in $\mathbb{Z_{p}} \times \mathbb{Z}_{q}$

If $p,q$ are distinct primes, it is true that the subset $\mathbb{Z} \times \mathbb{Z}$ is dense in $\mathbb{Z}_{p} \times \mathbb{Z}_{q}$. However, is it true that $\lbrace (x,x), x\in \mathbb{Z} ...
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22 views

about norm and p - adic number

For $x \in {\mathbb{C}_{p}}$ and $x \neq 1$ and $x^{p}=1$ with $p$ is a prime number. Then can we calculate $ {\left |x-1 \right |}_{p} ?$ We knew that ${\left | .\right |}_p$ is p -adic norm on ...
3
votes
1answer
57 views

Metric on $\mathbb{Q}$ for which the completion of $\mathbb{Q}$ is isomorphic to $\mathbb{Q}_2 \times \mathbb{Q}_3$. [closed]

Is there a metric on $\mathbb{Q}$ for which the completion of $\mathbb{Q}$ is isomorphic to $\mathbb{Q}_2 \times \mathbb{Q}_3$?
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0answers
29 views

Creating a grid of coloured points

I wish to create a grid of $31\times31$ points with coordinates of the form $(\frac k{30};\frac n{30})$ within the unit square, and give each point a colour based on the 2-adic value of both its ...
2
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1answer
70 views

Structure of $\Bbb Q_p/\Bbb Q$

The group $\Bbb A_\Bbb Q/\Bbb Q$ of adeles mod rationals is: Isomorphic to the solenoid Isomorphic to the group $\Bbb A_\Bbb Z/\Bbb Z$ Dual to the rational numbers This raises the question of the ...
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0answers
34 views

Error when tensoring with $p$-adic integers

It must be simple but I cannot find the error in the reasoning. Let $p$ be a prime number, we know that $\mathbb Z_p$ is flat over $\mathbb Z$ so that we can take the tensor product of the exact ...
2
votes
1answer
56 views

“Prime decomposition” for the profinite integers, adeles, or p-adics?

The strictly positive integers can be decomposed into a product of prime powers. Likewise, the positive rationals can be decomposed into a product of (possibly negative) prime powers. Another way to ...
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0answers
26 views

Is a locally constant function from a profinite abelian p-group constant modulo an open normal subgroup?

Suppose $f$ is a p-adic valued function from a profinite abelian group $B$ ($Z_{p}$ or $Z_{p}^{*}$) such that for all $x$ in $B$ there is an open set $N(x)$ around $x$ such that $f$ is constant on ...