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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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 $d$, $ f=(...
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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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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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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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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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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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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 ${\...
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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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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 ...
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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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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 ...
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“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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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 $N(...
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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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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 $$R=\{x \in k : |x| \leq 1\}$$...
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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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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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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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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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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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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 two, ...
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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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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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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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$\mathbb{Z}_p$ 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 ...
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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 ...
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92 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 ...
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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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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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$2$-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 ...
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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 $\...
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$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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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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$p^a\mid f(v) \implies p^a\mid f(w)$ 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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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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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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62 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 ...
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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$$(1 + x)^a = \sum_{n=0}^\infty \binom{a}{n} x^n.$$...
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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 $|xy|_2=|x|...
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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 ...
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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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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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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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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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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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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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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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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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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 of ...