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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Exactly 50% of 2-adic integers have the norm equal to $1$, exactly 25% have the norm $\frac{1}{2}$ and so on?

The question itself is more general and relates to all p-adic numbers, but it's really easy to show the principle using 2-adics. The definition of p-adic norm in most textbooks is not easy to ...
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66 views

Intuition behind Hensel's Lemma

In an Elliptic Curves course, my lecturer states Hensel's Lemma as the following: Let $k$ is a field that is complete with respect to a non-archimedian norm $|.|$ and $$R=\{x \in k : |x| \leq ...
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31 views

How does one visualize elements of standard Iwasawa algebra?

How does one think of elements of standard Iwasawa algebra of a profinite abelian group $B$ ? I mean like elements of $Z_{p}$ are represented as infinite series, is there a way to think of elements of ...
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131 views

Is $1+T$ a topological generator for $Z_{p}[[T]]$?

Is $1+T$ a topological generator for $Z_{p}[[T]]$? ($Z_p$ is the ring of p-adic integers)
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43 views

equivalent norms on vector space

For $(F,\left \| . \right \|)$ is a field with local compact norm and $V$ is a vector space over F. Can we have all norms on $V$ are equivalent?
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77 views

Fundamental unit of real quadratic field

Let $K=\mathbb{Q}(\sqrt d)$, where $d$ is squarefree and greater than $1$. Assume that $5|(d+1)$ or $5|(d-1)$. Let $\mathfrak{P}$ be a prime of $K$ over $5$. Let $\delta _o$ be a fundamental unit of ...
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1answer
78 views

P-adic numbers and infinity? Does infinity as a limit exist for p-adics?

I don't think I understand how p-adic numbers relate to the usual concept of infinity. The wiki page and various sources on the internet did not help. Let's see for example the 10-adic counterparts ...
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46 views

Distinct zeros of polynomials in $5$-adics.

How many distinct zeros does each of the following polynomials have in $\mathbb{Z}_5$? $f(x) = x^3 + 5x + 5$; $g(x) = x^5 + 2$; $h(x, y) = x^2 + y^2$. I know how to do the first ...
3
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1answer
48 views

zeros of p-adic power series

Suppose I have $f(x) \neq 0 \in \mathbb{Z}_p[[x]].$ If it can help it is of the form $g(x^p)+ph(x)$. Under which conditions can I say that $f(x)$ has finitely many zeros in $\bar{\mathbb{Q}}_p$? ...
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110 views

Understanding $Gal(\bar k /k)$

According to this book, An Introduction to the Langlands Program, One of the fundamental goals of modern number theory is to understand the Galois group $Gal(\bar k /k)$ where $k$ is a local or a ...
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1answer
27 views

Condition inverse $p$-adic number

Take $p$ prime, $n \in \mathbb{Z}_{>0}$ and $x \in \mathbb{Z}_p$. Suppose that $p$ isn't a divisor of $$x = (x_j + p^j \mathbb{Z})_{j \in \mathbb{Z}_{>0}},$$ then one can prove that the first ...
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101 views

$\mathbb{Z}_p$ is a complete metric space in which $\mathbb{Z}$ is dense

I don't quite understand a few parts of the proof of proposition $3$. What is meant by "the ideals $p^n\mathbb{Z}_p$ form a basis of neighborhoods of $0$"? After reading the definition of a ...
2
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1answer
49 views

Let $p$ and $q$ be distinct primes. Can you prove the sequence $\{p^n\}_{n \in \mathbb{N}}$ is not Cauchy under the given metric on $\mathbb{Q}$?

This is an elementary $p$-adic theory question. Granted $d(x,y)=|x-y|_q$ is a metric on $\mathbb{Q}$, and $|\cdot|_q$ is a norm such that $$|x|_q=q^{-ord_q x}$$ where $ord_q x$ is the largest ...
2
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1answer
84 views

profinite completion of a ring of integers in a global field

It is well known that the profinite completion of the integers, $\hat{\mathbb{Z}} = \varprojlim \mathbb{Z}/n\mathbb{Z}$ is, by the Chinese remainder theorem, isomorphic to $\prod_p \mathbb{Z}_p$. I ...
2
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1answer
52 views

The topology of $\mathbb{Z}_p$

I don't know much about topology, but anyway... Assuming $\displaystyle\prod{A_n} =\prod_{n\geq 1}{A_n}$, why is $\mathbb{Z}_p$ closed in a product of compact spaces? Googling I found Tychonoff's ...
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1answer
46 views

Is there an isomorphism $\mathbb{C}_p \cong \mathbb{C}_q$ for primes $p \neq q$?

Let $p \neq q$ be distinct primes. Is there a field isomorphism $\mathbb{C}_p \cong \mathbb{C}_q$? Is there a field isomorphism $\mathbb{C}_p \cong \mathbb{C}$? If such an isomorphism exists, given ...
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96 views

$2$-adic sequence converging to $\sqrt{-7}$.

I am trying to construct a sequence in $\mathbb Q_2$ that is formed of rational numbers and converges to $\sqrt{-7}$, to prove that $(\mathbb Q, |\cdot|_2)$ is not complete. My lecturer stated that ...
4
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1answer
171 views

Is the real number $\sqrt{6}$ in $\mathbb{R}$ equal to the 5-adic number $\sqrt{6}$ in $\mathbb{Q}_5$?

My question is as in the title. That is, consider solving the equation $x^2-6=0$ in $\mathbb{R}$ and in the 5-adic field $\mathbb{Q}_5$ respectively. We obtain one $\sqrt{6}\in\mathbb{R}$ and one ...
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26 views

$p$-divisibility and $q$-th roots of unity

Let $p$ be a prime, $n$ a positive integer and $q=p^n$ a $p$- power. Let $w,v$ is a root of the $q,p$-th cyclotomic polynomial $Q_q,Q_p$, respectively. Let $f$ be a polynomial with coefficient in ...
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2answers
78 views

The algebraic closure of $\mathbb{Q}_p$

I am trying to explain why $\mathbb{Q}_p^{\text{alg cl}}$ is an infinite field extension of $\mathbb{Q}_p$ (unlike $\mathbb{C}/\mathbb{R}$ which has deg 2). Does the following argument work out... ...
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1answer
48 views

$p^a\mid f(v) \implies p^a\mid f(w)$ in $\mathbb Z[w]$

Let $p$ be a prime, $n$ a positive integer and $q=p^n$ a $p$- power. Let $w,v$ is a root of the $q,p$-th cyclotomic polynomial $Q_q,Q_p$, respectively. Let $f$ be a polynomial with coefficient in ...
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81 views

Unicity in $p$-adic Weierstrass Preparation Theorem: dividing both sides of an equation of formal power series

I would like to see whether the following proof is correct or not. Let $\Omega$ be a completion of an algebraic closure of $\mathbb{Q}_p$. Let $g$ and $g_1$ be power series in $1+X\Omega[[X]]$ that ...
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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 ...
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31 views

What to call a polynomial that with no roots in $\mathbb{Q}$ but does in $p$-adics

As an example, consider the polynomial $f(x)=x^2-2$. This polynomial has no roots over the rationals $\mathbb{Q}$. However, $\sqrt{2} \in \mathbb{Q}_7$, so it does have roots in the 7-adics. Is there ...
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1answer
59 views

$p$-adic Euler's totient function

Let $p$ be a prime number, $\overline{\mathbb{Q}_p}$ a fixed algebraic closure of $\mathbb{Q}_p$. Let $\alpha$ be a primitive $n$-th root of unity in $\overline{\mathbb{Q}_p}$ and $d$ its degree over ...
7
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2answers
110 views

Existence of $\sqrt{-1}$ in $5$-adics, show resulting sum is convergent.

I know that to prove the existence of a square root of $-1$ in $\mathbb{Z}_5$, I can just plug $x = -5$ and $a = 1/2$ into the Taylor expansion$$(1 + x)^a = \sum_{n=0}^\infty \binom{a}{n} ...
5
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1answer
190 views

Extending 2-adic valuation to real numbers

When proving Monsky's theorem, one of the steps, which, from what I have so far seen, no proof can avoid, is extending the 2-adic valuation to all real numbers, so that it still satisfies ...
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1answer
179 views

What is the Galois group of $\mathbb{Q}_p[\zeta] / \mathbb{Q}_p$, where $\zeta$ is a $p^r$th root of unity?

Let $\zeta$ be a $p^r$-th root of unity and $\mathbb{Q}_p$ the p-adic numbers. I would like to know the Galois automorphisms of $\mathbb{Q}_p[\zeta] / \mathbb{Q}_p$. I already know, the degree of ...
3
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1answer
80 views

converging power series over $p$-adic integers is a UFD

Denote by $|\cdot |$ the $p$-adic norm, and let $$\mathbb Z_p \{z\}=\left\{\sum_{n=0}^\infty a_nz^n;\ a_n\in \mathbb Z_p ,\ |a_n|\underset {n\rightarrow \infty} {\longrightarrow 0} \right\}$$ the ...
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43 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 ...
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27 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 $U$ ...
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1answer
46 views

Converges to 0 vs. Diverges to 0; Terminology in p-adic Analysis

In practice, in p-adic analysis, when referring to a sequence of numbers it is common to use the terminology "converges to 0". However, isn't this terminology, technically, incorrect by the definition ...
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123 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 ...
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80 views

The p-adic expansion of 1/2 for odd p

I just started self-studying p-adic analysis, using Alain M. Robert's book A Course in p-adic Analysis. So, I wanted to make sure that I got this correct before continuing and developing bad habits ...
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1answer
54 views

Non real complex in metric completions of $\mathbb Q$

Process of completion of $\mathbb Q$ using the absolute value $|x|$ does not touch to the non-real complex numbers which are added to $\mathbb Q$ via extensions fields. However completion of $\mathbb ...
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1answer
50 views

Roots of $p$-adic irreducible polynomials

A finite field $\mathbb{F}_p$ posesses the property that for any irreducible polynomial $f\in\mathbb{F}_p[x]$ adjoining any root of $f$ automatically adjoins all roots of $f$. (In other words, any ...
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2answers
82 views

Reference for the p-adic numbers

Can anyone give me a reference (book or a paper) that introduces the p-adic numbers and their important properties? Also, I would love if that reference contained some not to advanced applications ...
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2answers
44 views

Error in or another way of calculating $\frac{1}{2} \in \mathbb{Q}_3$

I want to find the $p$-adic expansion of $\frac{1}{2} \in \mathbb{Q}_3$. I begin with noting that $- \frac{1}{2} = \frac{1}{1-3} = \sum_{n=0}^{\infty} 3^n$. Therefore, $\frac{1}{2} = 1 - \frac{1}{2} = ...
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3answers
118 views

Understanding $p$-adic fields

OK, I'm completely lost on this. Define the $p$-adic integers $\mathbb{Z}_p$ as the projective limit $$\lim_{\leftarrow} \mathbb{Z}/p^n \mathbb{Z}.$$ So, if $a \in \mathbb{Z}_p$, then $a$ can be ...
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1answer
53 views

Proof that $\mathbb Q_p$ is unique up to unique isomorphism preserving the absolute values

On pages 58-59 of Gouvea's $p-$adic Numbers: An Introduction, he gives the following proof that the field $\mathbb Q_p$, constructed using equivalence classes of Cauchy sequences, is unique up to ...
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Primitive $p^n$-th root of unity in $\bar{\mathbb{Q}}_p$.

I am trying to solve the following exercise in Koblitz's "$p$-adic Numbers, $p$-adic analysis, and Zeta-Functions". Let $p$ be a prime. Let $a$ be a primitive $p^n$-th root of unity in ...
2
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1answer
59 views

Solving equations in $\mathbb{Z}_3$ with Hensel's Lemma

Further to the post here, I'm trying to find the $n \in \mathbb{Z}$ such that there is a solution to the equation $$ x^3 +3x+y^3+3y=n$$ in $\mathbb{Z}_3$. Now, I've been able to show that in the ...
3
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1answer
33 views

Valuation of a particular element

I am tying to compute the valuation of a particular element of $\mathbb{Q}_p$. I am trying to compute $\operatorname{val}_p(P)$ where $P=\frac{\log(1+p^2)}{\log(1+p)}$ and $\log$ is the $p$-adic ...
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39 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 ...
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66 views

$p$-Adic complex numbers, expansion into series

For any field of $p$-adic numbers $\mathbb{Q}_p$ there is a unique, up to an isomorphism, extension $\mathbb{C}_p$ that is both algebraically and metrically complete, the field of $p$-adic complex ...
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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 $R$ ...
2
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2answers
47 views

$p$(ain)-adic number sequence

I am trying to figure out how $p$-adic numbers work and currently am having trouble wrapping my head around how they work, so I made a pun! HAH! Jokes aside, I am working on this question Show ...
4
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1answer
52 views

Show an intersection of Galois groups is trivial

Let $L/K$ be a finite abelian extension of number fields, and for an extension of places $w/v$ consider the local Artin map $\Phi: K_v^{\ast} \rightarrow Gal(L_w/K_v)$, defined via the global Artin ...
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38 views

Is the $\mathbb{Q}_2$- Space$ (\mathbb{Q}_2[\zeta], trace(cxy))$ hyperbolic?

I am working at a Problem for some time and it comes down to the question: Let $K:=\mathbb{Q}_2[\zeta]$ be a cyclotomic extention of the dyadic field $\mathbb{Q}_2$. For any $c \in ...
2
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1answer
32 views

Proof that an archimedean absolute value on $\mathbb Q$ is equivalent to $|\ |_\infty$

I have included a bolded comment in a step in the part of Gouvea's proof of Ostrowski's theorem where he shows that an archimedean absolute value on $\mathbb Q$ is equivalent to $|\ |_\infty$ (the ...