# Tagged Questions

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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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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### 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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### 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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### 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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### 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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### 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 $\mathbb{Q_7}$
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 $P_K = \{ x\in ... 1answer 134 views ### Inverse mapping theorem over a complete non-archimedean field Let$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}$... 1answer 244 views ### About$p$-adic Topology on$\mathbb{Z}$I want some notes about$p$-adic Topology and some properties with proves about$\mathbb{Z}$with this Topology. Thank you. 1answer 104 views ### Why$\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$linear (and its topology a linear ... 1answer 869 views ### Open set = *disjoint* union of open balls? Recently, i have read the assertion that in$Q_p$, the p-adics, every open set is a disjoint union of open balls. This is not true for a general metric space, see for example How to make open covers ... 1answer 117 views ### Reference requested: 'decomposition' of Haar measure on the adeles. Since the adeles$\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 ... 0answers 44 views ### 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 ... 1answer 335 views ### 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 ... 0answers 92 views ### 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 $$\int_{C}h(u)|u|d^{*}u- ... 1answer 254 views ### Local solutions of a Diophantine equation I am trying to prove that the equation$$3x^3 + 4y^3 +5z^3 \equiv 0 \pmod{p}$$has a non-trivial solution for all primes p. I am sure that this is a standard exercise, and I have done the easy ... 1answer 103 views ### Unramified extension is normal if it has normal residue class extension Let K/F be an unramified extension such that \rho_K / \rho_F (the corresponding extension of residue classes) is normal. Prove K/F is normal. I guess I need to do some polynomial lifting, but ... 2answers 378 views ### Question about p-adic numbers and p-adic integers I've been trying to understand what p-adic numbers and p-adic integers are today. Can you tell me if I have it right? Thanks. Let p be a prime. Then we define the ring of p-adic integers to ... 1answer 301 views ### Are all p-adic number systems the same? After just having learned about p-adic numbers I've now got another question which I can't figure out from the Wikipedia page. As far as I understand, the p-adic numbers are basically completing ... 1answer 148 views ### A property of non-Archimedean metrics I have recently been reading about non-Archimedean metrics on fields (in Koblitz: p-adic Numbers, p-adic Analysis, and Zeta-Functions), and came across the exercise: Prove that a norm \|.\| on ... 1answer 125 views ### diagonalizing a matrix over the \ell-adics Let M be a 2 \times 2 matrix with coefficients in \mathbb{Z}_{\ell} whose characteristical polynomial is$$ P(T) = T^2- (a+d) T + (ad-bc). $$I've encountered the following assertion: If ... 4answers 424 views ### 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 ... 1answer 110 views ### 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 ... 0answers 127 views ### 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 = ... 2answers 442 views ### 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 ... 2answers 274 views ### 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 ... 1answer 110 views ### 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 ... 1answer 185 views ### 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, ... 4answers 329 views ### 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 ... 2answers 300 views ### 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 ... 1answer 283 views ### 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 ...
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. ...