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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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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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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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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127 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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$p$-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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45 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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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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35 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} ...
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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 ...
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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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43 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 ...
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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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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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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 ...
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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 ...
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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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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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$p$-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 $R$ ...
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2answers
40 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 ...
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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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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 ...
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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 ...
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Squares in $\mathbb Z_p$

Let $p\neq 2$, then I want to understand that every element in $\mathbb Z_p$ (p-adic integers) is a square. For the prove one must see that $2$ is invertible in $\mathbb Z_p$. But $2$ is the element ...
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Roadmap to $p$-adic numbers: where a self-learner should look for references

TL;DR at the end of the question. I’m currently trying to learn as much as possible about p-adic numbers. I’m not sure what is the most fascinating part of the theory, but the use of the adjective ...
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Fermat's little theorem for p-adic integers

Just a short question: Is Fermat's little theorem applicable in the p-adic integers $\mathbb Z_p$? If yes, why?
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Structure of the group $\{1+p\mathbb Z_p \}$

In preparation for algebraic number theory I am reading Serre : A course in Arithmetic. I stuck in understanding a proof (p.17): Notation: $U_n=1+p^n\mathbb Z_p$ Actually there are many things ...
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The group $(1+p\mathbb Z_p)/(1+p^{n}\mathbb Z_p)$

I want know some information about the group \begin{equation*} \frac{(1+p\mathbb Z_p)}{(1+p^{n}\mathbb Z_p)} \end{equation*} (the Quotient group). What is the order of this group? I guess $p^{n-1}$ ...
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Which properties determine the uniqueness of the local Artin map?

Any abelian extension of local fields can be realized as the completion of a global abelian extension. So let $L/K$ be abelian, $w/v$ an extension of places. From the global Artin map on ideles we ...
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In $\mathbb Q_p$, proving every open ball is the disjoint union of more than one open ball

I'm reading the Foundations chapter of Gouvea's p-adic Numbers: An Introduction, and I'm trying to solve the following problem he poses to the reader: Take the $p-$adic absolute value on $\mathbb ...
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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 $\mathfrak p$ ...
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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 = ...