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 liftings on SAGE

I asked a question the other day: Multidimensional Hensel lifting which @Hurkyl kindly and very elegantly answered. A follow-on from this is that I have tried to implement exactly the "algorithm" ...
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Profinite and p-adic interpolation of Fibonacci numbers

On the topic of profinite integers $\hat{\bf Z}$ and Fibonacci numbers $F_n$, Lenstra says (here & here) For each profinite integer $s$, one can in a natural way define the $s$th Fibonacci ...
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Multidimensional Hensel lifting

I have a question about a practical application of (some) generalised form of Hensel's Lemma. I cannot find it stated in an appropriate form in Bourbaki or anywhere else, so here goes ... Let $p$ be ...
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p-adic numbers and $\mathbb{F}_p$

As you probably know, there is a morphism of rings : $\mathbb{Z}_p\longrightarrow \mathbb{Z}/p\mathbb{Z}$ which sends a formal sum $\sum_{i\geq 0}a_ip^i$ to $a_0$ (here $\mathbb{Z}_p$ is the ring of ...
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201 views

Roots of unity in $\mathbb Q_p$

Is there a way to get the number of solutions of an equation like $x^n=1$ in the p-adic field $\mathbb Q_p$, where $p$ is a prime and $n$ a positive integer? I know that for $n=p-1$ there are $p-1$ ...
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If $(x_n)_{n\geq 1}$, $x_n \in \mathbb{Q}$ is p-adic Cauchy, show ord$_p(x_n)$ eventually constant.

Number Theory 1: Fermat's Dream asks the reader to verify the following. They then use this to extend the definition of Ord$_p$ to $\mathbb{Q}_p$. Let $p$ be prime and $a\neq 0$. If ...
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What is the index of the $p$-th power of $\mathbb Q_p$ in $\mathbb Q_p$

In this book it is listed as an exercise to compute the index $[\mathbb Q_p:\mathbb Q_p^p]$. This exercise is appended to a section concerning the structure of unit-group filters, investigating some ...
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Origin and use of an identity of formal power series: $\det(1 - \psi T) = \exp \left(-\sum_{s=1}^{\infty} \text{Tr}(\psi^{s})T^{s}/s\right)$

The following is a historical question, but first some background: Let $\psi$ be a linear operator from a vector space to itself. The following two expressions, viewed as formal power series, can be ...
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Set of locations where the Hilbert symbol is not equal to $1$

Let $V$ be the set of prime together with the symbol $\infty$. For a prime $v=p$, denote the $p$-adic numbers by $\mathbb{Q}_p$ and the real numbers by $\mathbb{Q}_\infty$. For $v\in V$ the Hilbert ...
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When a number is a square in the p-adic rationals - proof question (Quadratic Residues)

I'm a little stuck with the proof of a theorem I'm trying to understand. The theorem is as follows: "For odd prime $p$, suppose for $\alpha \in Q_{p}$ (the p-adic rationals) that $|\alpha|_p=1$. Then ...
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Preservation of being a norm under field extension

I'm reading a paper that purports to prove the proposition: Let $K/E$ be a cyclic extension of CM number fields of degree p (an odd prime number). Let $G$ be the Galois group. Let $t$ be the number ...
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98 views

Valuation rings of complete non-archimedean fields which are not local

I would like to know how are the valuation rings of complete non-archimedean fields which are not local. Take, for example, $\mathbb{C}_p$, and its valuation ring, ...
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233 views

Roots of 1 in $\mathbb Q_p$

How to prove, that all roots of 1 in $\mathbb Q_p$ are roots of $x^{p-1}-1$? If we consider the ring homomorphism $$ \mathbb Z_p \to \mathbb F_p^*, $$ then we see, that all the roots in power $p-1$ ...
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A $p$-adic “Jacobi sum” with an unramified character

Working over a $p$-adic field with absolute value $|\cdot|$, let $\chi$ be a character on ${\mathfrak o}^\times$ with conductor $n\ge 1$, meaning that $n$ is the smallest integer such that $\chi$ is ...
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Are the p-adic integers the ring of integers of the field of p-adic numbers?

This question was much simpler, but as I was typing it, it became a chain of questions. My starting question was Is $\mathbb{Z}_p$ (obtained by the inverse limit procedure with the directed ...
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How do you take the multiplicative inverse of a p-adic number?

I am reading the wiki page for p-adic numbers and it states that they are a field extension of the rationals so each member has to have a modular multiplicative inverse. So how would I take the ...
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$p$-adic unit for $p\neq 2$ is a square in $\mathbb{Z}_p$ if and only if its first digit is quadratic residue modulo $p$.

I heard the statement from the title of this question. $\newcommand{\of}[1]{\left(#1\right)}$ $\newcommand{\df}{\mathrel{\mathop:}=}$ So if I am not totally confused, this formalizes to: Let ...
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Quasi-linear time fully homomorphic encryption using p-adic ring homomorphism

I recently encountered a breakthrough in FHE crypto, which claims to have a literally quasi-linear time FHE without any "lambda" factor in the keys and no noise in the cipher-text. This fully ...
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132 views

Cyclotomy in extensions of $\mathbb{Q}_p$

Let $p$ be a prime number and $K$ be a finite extension of $\mathbb{Q}_p$. Denote by $\zeta_{p^n}$ a primitive $p^n$-th root of unity (where $n$ is a positive integer). Assume that $K$ contains ...
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hensel's lemma for the prime 2

Hensel's lemma for odd primes, as I understand, allows you to determine solutions to various homogeneous quadratic equations modulo $p^n$ by solving an appropriate equation mod $p$ and lifting up. ...
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Powers of $\Bbb Z_p\cap \Bbb Q$ on $1+x\Bbb F_q[[x]]$

Set $q=p^r$ for the finite field $\Bbb F_q$. In the formal power series ring $\Bbb F_q[[x]]$, there is a notion of convergence given by the underlying $(x)$-adic topology. If ...
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Galois group of maximal $p$-extension of $\mathbb{Q}_l(\zeta_p)$?

Let $F$ be the field obtained by adjoining to $\mathbb{Q}_l$ a $p$-th root of unity, with $p \not = l$. Denote by $F(p)$ the maximal $p$-extension of $F$, i.e. the maximal extension $L:F$ such that ...
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Show that $x^4+1$ is reducible in p-adic numbers $\mathbb{Q}_p$ for p>2 prime.

This is a homework problem for algebraic number theory but I'm having trouble getting started. Do I use induction in general, or show this holds for $p \equiv 1,3$ (mod 4)? Any help would be ...
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255 views

continued fraction expression for $\sqrt{2}$ in $\mathbb{Q_7}$

Hensel's lemma implies that $\sqrt{2}\in\mathbb{Q_7}$. Find a continued fraction expression for $\sqrt{2}$ in $\mathbb{Q_7}$
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Structure of $\mathbb{Q}_p(\zeta_p)$

Let $p \ne 2$ be prime number and denote by $\zeta_p$ the p-th root of unity. It's well known that $K = \mathbb{Q}_p(\zeta_p)$ has $t=1 - \zeta$ as prime element (generator of the Ideal $P_K = \{ x\in ...
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Inverse mapping theorem over a complete non-archimedean field

Let $K$ be a complete field with respect to a non-trivial non-archimedean absolute value $|\cdot|$. Let $E$ be a vector space over $K$. A norm $||\cdot||$ on $E$ is a map $E \rightarrow \mathbb{R}$ ...
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244 views

About $p$-adic Topology on $\mathbb{Z}$

I want some notes about $p$-adic Topology and some properties with proves about $\mathbb{Z}$ with this Topology. Thank you.
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104 views

Why $\mathbb{Z}$ with $p$-adic topology is precompact?

Why $\mathbb{Z}$ (group of integer numbers) with $p$-adic topology is a countable precompact metric group with a linear topology? Note : Call a topological group $G$ linear (and its topology a linear ...
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Open set = *disjoint* union of open balls?

Recently, i have read the assertion that in $Q_p$, the p-adics, every open set is a disjoint union of open balls. This is not true for a general metric space, see for example How to make open covers ...
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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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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 ...
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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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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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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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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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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 ...
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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 ...
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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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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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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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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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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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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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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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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, ...
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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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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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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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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. ...