Questions tagged [p-adic-numbers]
The p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems
272 questions
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Minimal odd $k$ for stabilization of solutions of $y^k=y$ in $\mathbb{Z}_n$
We consider fixed-point equations of the form $$y^k = y \qquad (k\in\mathbb{N})$$ in $\mathbb{Z}_n$, where $\mathbb{Z}_n$ denotes the ring of $n$-adic integers (i.e., the projective limit $\mathbb{Z}...
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Validity of multiplying a congruence and its modulus by a constant factor [migrated]
I am a student currently in a dispute regarding a step in a proof. I have the following congruence involving a fraction:$$\frac{x(x+1)}{2} \equiv \frac{y(y+1)}{2} \pmod{2^n}$$In my proof, I performed ...
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Predicting the smallest nontrivial natural number whose congruence speed never stabilizes in a given numeral system
[This question uses the same nomenclature already provided in https://mathoverflow.net/q/504298].
Let $r > 1$ and $a > 1$ be integers, and consider the radix-$r$ numeral system.
For each integer ...
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$p$-adic concepts having no analogue in the real case
I've always been fascinated how among completions of $\mathbb{Q}$, the real field $\mathbb{R}$, althrough historically more "ancient", seems to be the odd one out. Many interesting concepts ...
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Accumulation points of $(3^n)$ in ${\bf Z}_2$
The sequence $(2^n)_{n\in {\bf N}}$ converges to $0$ in the compact ring of dyadic integers ${\bf Z}_2$.
What is the set of accumulation points of the sequence $(3^n)_{n \in {\bf N}}$ in ${\bf Z}_2$?
...
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How do I write down an explicit primitive 9th root of unity in the invariant 1/3 central division algebra over $Q_3(\zeta_3)$?
Here is a very naive question about division algebras. Let $K$ denote the $p$-adic number field given by adjoining a primitive third root of unity to $\mathbb{Q}_3$. Let $D$ denote the central ...
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Isometric map of affinoid p-adic algebras
Let $A = \mathbb{Q}_p\langle t_1, \dots, t_n \rangle = \mathbb{Q}_p\langle T_1, \dots, T_n \rangle/J$ be an $p$-adic affinoid algebra generated by $t_1, \dots, t_n$ with its norm being the quotient ...
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Does $\text{Gal}(K(F[\pi^{\infty}])/K \cong \text{Gal}(K(G[\pi^{\infty}])/K$ imply $F[\pi^n] \cong G[\pi^n]$?
Let $K$ be a $p$-adic number field with ring of integers $\mathcal{O}_K$ and uniformizer $\pi$. Let $F$ and $G$ be two $d$-dimensional formal groups of height $h$ over $\mathcal{O}_K$. Denote:
\begin{...
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Orthogonalization of quadratic forms over a $p$-adic Banach space
Let $X$ be an arbitrary set. Let $H = c_0(X, \mathbb{Q}_p)$ be the $p$-adic Banach space with sup norm. Let $\langle \cdot, \cdot \rangle$ be a symmetric, nondegenerate $\mathbb{Q}_p$-bilinear form on ...
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$v$-adic expansions of non-$p$th powers in global fields
Let $k$ be a global function field of positive characteristic $p$ (e.g. $k = \mathbb{F}_p[t]$). Let $x \in k$ be non-zero and assume that $x$ is not a $p$th power.
For each place $v$ of $k$, we can ...
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Monsky's theorem in 3d
Monsky's theorem, which has a rather fancy proof, states that it is impossible to triangulate a square into an odd number of triangles of the same area.
I am interested to find out whether the 3-...
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On the $p$-adic valuations of coefficients in holonomic-like sequences of vectors
The following problem arises after a number of preprocessing steps from studying how many zeroes certain holonomic sequences can have.
Fix a prime $p$, a natural number $k$ and two non-zero natural ...
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If a $p$-adic power series vanishes at $\zeta_{p^n}^a-1$ for all $n,a$, is it divisible by $\log(1+T)$?
Let $p$ be a prime number, and let $H(\mathbb{C}_p)$ denote the ring of power series $f(T)\in \mathbb{C}_p[[T]]$ such that $f(T)$ converges in an open ball of radius $1$ about $0$. n.b. that this is ...
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On the integral closure of $\mathbb{Z}_p$ over a $p$-adic cyclotomic field [duplicate]
Notation. Let $p$ be a prime. Let $\mathbb{Q}_p$ be the $p$-adic number field and let $\mathbb{Z}_p$ be the ring of $p$-adic integers. For any positive integer $n\ge2$, let $\zeta_n\in\mathbb{Q}_p^{{\...
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On $a_n(x)=\sum_{i,j=0}^n \binom ni^2\binom nj^2\binom{i+j}ix^{i+j}$ (III)
As in Question 491655 and Question 491762, we define
$$a_n(x):=\sum_{i,j=0}^n\binom ni^2\binom nj^2\binom{i+j}ix^{i+j}$$
for each nonnegative integer $n$.
Here we pose some curious congruences ...
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Preperiod of powers of matrices modulo m
Let $A$ be a square matrix with integer entries and let $m$ be a positive integer. From the pigeonhole principle it follows easily that the sequence
$$I,A, A^2, A^3,\; \dots \pmod m$$
is eventually ...
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Fractional field of Witt ring of algebraic closure
So let $k$ be a perfect field of characteristic $p$. Let $W(\cdot): \mathtt{Rings}\rightarrow \mathtt{Rings}$ denote the functor taking rings to their corresponding Witt rings. Denote $K_0 = \...
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Why is $vv^T+\mathbb{I}_{n-1}$ congruent to a diagonal matrix over $p$-adic integers $\mathbb{Z}_p$ when $p≠2$?
Let $ v \in \mathbb{Z}_p^{n-1} $, and consider the symmetric matrix
$$
A = v v^T + I_{n-1}
$$
over the ring of $p$ -adic integers $ \mathbb{Z}_p $.
Why is this matrix congruent (i.e., there exists $U ...
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Newton Polygon properties of $f(x+a)$, when the Newton polygon for $f(x)$ is known
Let us consider a polynomial $f(x) = x^n + f_{n-1}x^{n-1} + \cdots + f_0 \in \mathbb{Z}_p$, where $\mathbb{Z}_p$ is set of all $p-$adic integers. Let $-s/r$ be the slope of Newton Polygon of $f(x)$. ...
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Matrices over $\mathbb{Z}_l$
Suppose I have a matrix over the $l$-adic integers $\mathbb{Z}_l$ which is diagonalizable over $\mathbb{Q}_l$. How to classify such matrices by similarity over $\mathbb{Z}_l$?
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Index of subgroups of norm-1 elements in local field extensions
Let $F$ be a local field and let $ E = F[\sqrt{\epsilon}] $ denote the quadratic unramified extension of $ F $. Let $ P_E $ be the unique maximal ideal of the ring of integers $ O_E $ in $ E $, and ...
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135
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Levis, parabolics and Bruhat-Tits over Henselian local rings
Let $(R,m)$ be a Henselian local ring with algebraically closed or finite residue field $k$ and fraction field $F$. For example, we may work with $R=W(\mathbb F_p^{alg})$.
The paper "Reductive ...
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Is there a theory of completions of semirings similar to $I$-adic completions of rings?
Let $L = \text{Con } (\mathbb{N}, 0, +) \setminus \Delta$ be the lattice of monoid congruences on the naturals, excluding the trivial congruence. As it happens, every $\theta \in L$ is the meet of ...
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1
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Can a p-adic ball cover a p-adic ball?
Are there a polynomials $f_1,...f_n \in \mathbb{Z}_p[x_1,...x_n]$ with there coeficients $p$-adic integers s.t.
A map $F:\mathbb{Z}_p^n\rightarrow \mathbb{Z}_p^n$ defined by $f_1,...f_n$
satisfy the ...
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1
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The simply connectedness of $\mathbb{A}^n_{\mathbb{Q}_p}$
My question is how to prove the affine $n$-space over $p$-adic number $\mathbb{Q}_p$ is simply connected.
To be precise,
Let $X$ be $p$-adically analytic manifold, $f:X\rightarrow \mathbb{A}^n_{\...
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1
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A specific $2$-dimensional Galois representation of $G_{\mathbb{Q}_2}$ and its Langlands correspondence
I am interested in understanding a situation in (classical, not $p$-adic) local Langlands for $\mathrm{GL}_p(\mathbb{Q}_p)$. An example of
it is as follows: Let $F=\mathbb{Q}_2$ and $E$ be the ...
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Galois action on the cohomology of a curve over a local field with bad reduction
Let $C/\mathbb Q_p$ (or a p-adic local field more generally) be a smooth projective curve with split semistable reduction over $\mathbb Z_p$. What can we say about the action of the Galois group $\...
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A question on distinguished pairs
I am reading Alexandru, Popescu, and Zaharescu, "On the Closed Subfields of $\mathbb{C}_p$" (see https://tinyurl.com/kknmzbyx). The authors give the following definition:
Let $\alpha, \beta \...
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What is the action of the Galois group due to Lubin-Tate formal group on the respective Tate module?
It is a well-known fact that a Tate module $T_p(A)$ of an abelian group (abelian variety or commutative group scheme) $A$ over a field $K$, equipped with a continuous action of the respective absolute ...
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When is a p-adic number a $p$th power over the field it generates
Does there exist an $\alpha$ in an algebraic closure $\mathbb{Q}_p^{\rm alg}$ of $\mathbb{Q}_p$ such that $\frac{p}{p-1} \geq v(\alpha)>0$ and $1+\alpha$ is a $p$th power in $\mathbb{Q}_p(\alpha)$?...
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Ramification at particular level of a tower of extensions of local field
Let $K$ be an unramified extension of the $p$-adic number field $\mathbb{Q}_p$.
Suppose we have a tower of extensions:
$$K=:K(u_0) \subset K(u_1) \subset K(u_2) \subset K(u_3) \subset \cdots \subset ...
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1
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Finite-order automorphisms in the absolute Galois group of a $p$-adic field?
I'm searching for a sort of analogue of the complex conjugation.
More precisely, let $K$ be a characteristic zero field complete with respect to an ultrametric absolute value. Let $C$ be the ...
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Integral closure in the algebraic closure of $p$-adic numbers
Let $p$ be a prime number and let $\overline{\mathbb{Q}}_p$ be a fixed algebraic closure of the $p$-adic numbers $\mathbb{Q}_p$. It is well know that the ring of integers of $\mathbb{Q}_p$ is the ring ...
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Problem Deducing the value of Quadratic Hilbert Symbol from Explicit Formulas
This question concerns the explicit law for the Hilbert Symbol given in Sur les lois de réciprocfites explicites I by Henniart. I am trying to deduce the classical value of the Hilbert Symbol in $\...
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Does there exist a polynomial that extracts the highest digit of an integer in base p?
Given an odd prime $p$, a positive integer $1 \lt n$, and an integer $x \in \mathbb{Z}/p^n\mathbb{Z}$, does there exist an an integer-coefficient polynomial that extracts the highest digit of $x$?
The ...
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What circumstances guarantee a p-adic affine conjugacy map will be a rational function?
Let $\Bbb Q_p$ be a p-adic field and let any element $x$ of $\Bbb Q_p$ be associated with a unique element of $\Bbb Z_p$ via the quotient / equivalence relation $\forall n\in\Bbb Z:p^nx\sim x$
Then in ...
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Can every $\ast$-algebra be represented in this space of matrices?
Let $k$ be a field with characteristic $0$. For every set $X$, let $\mathcal{B}(X)$ be the set of (possibly infinite) matrices $T = (T_{x,y})_{x,y \in X}$ with coefficients in $k$ such that in each ...
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Can the p-adic be countable?
Recently arxiv submitted a new paper (Andrej Bauer, James E. Hanson, The Countable Reals) claiming an incredible theorem that Dedekind reals are not sequence-avoiding, and furthermore obtaining a ...
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Maximum modulus principle over the $p$-adic integers
Consider $\mathbb{Z}_p$ the $p$-adic integers. Let $f\in\mathbb{Z}_p[x]$ be an arbitrary polynomial in one variable. Write $f(x) = \sum_{k}a_kx^k$. Is it true that $\|f\|:= \max_k |a_k|_p = \sup_{t \...
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Time complexity of Magma's `NormEquation` for quadratic extensions of $2$-adic fields
Note: This is similar to, but easier than, a previous question I asked here. It is a different question! I'm hoping this one might get an answer because it concerns a standard algorithm, whereas the ...
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Subgroup of p-adic units
Let $\left(\widehat{\mathbb Z}\right)^\times=\prod_p{\mathbb Z}_p^\times$
be the unit group of the ring $\widehat{\mathbb{Z}}$, which is the profinite completion of $\mathbb Z$.
We give it the product ...
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Is the extension field by zeros of $x^{2m}-p^{2(m-1)}=0$ over $\mathbb Q_p$ totally ramified?
Let $m \geq 2$ be an integer. Consider the polynomial $f(x)=x^{2m}-p^{2(m-1)} \in \mathbb{Q}_p[x]$.
I want to study the field extension by the zeros of $f(x)$ over $\mathbb{Q}_p$.
What is the degree ...
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Computing preimage of element under norm map of quadratic extension of $2$-adic fields
Let $F$ be a $2$-adic field, i.e. a finite extension of the $2$-adic numbers $\mathbb{Q}_2$. Suppose that I have a quadratic extension $E = F(\sqrt{d})$ of $F$. Given a unit $\alpha \in \mathcal{O}_F^\...
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Bezout-type theorem for $p$-adic analytic plane curves
Let $p$ be a prime, and let $f,g \in \mathbb{Z}_p[[x,y]]$ be power series convergent on all of $\mathbb{Z}_p$. Suppose that the intersection of the analytic plane curves cut out by $f$ and $g$ is ...
3
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221
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Approximating $p$-adic power series by polynomials
Let $p$ be a prime, and let $f \in \mathbb{Z}_p[[x_1,\dots,x_d]]$ be a power series convergent on all of $\mathbb{Z}_p^d$. We make the following definition concerning the approximation of $f$ by ...
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2
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206
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How do you prove that the series 5, 25, 625, ... can be continued forever to give a 10-adic solution to $n^2=n$? [closed]
How do you prove that the series 5, 25, 625, ... can be continued forever to give a 10-adic solution to $n^2=n$? Here's a proof for a different solution (...1787109376): https://oeis.org/A018248/...
2
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215
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p-adic Banach space and complete tensor product
Let $p$ be a prime and $\mathbb{C}_{p}$ the completion of the algebraic closure of the $p$-adic number field $\mathbb{Q}_p$.
Let $M$ be a $\mathbb{Q}_p$-Banach space.
We denote by $M\mathbin{\widehat{\...
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1
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460
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Conductor and local Kronecker–Weber theorem
Given an abelian extension $K$ of $\mathbb{Q}$, the global Kronecker–Weber theorem tells us that there exist a positive integer $N$ and a primitive $N$-th root of unity $\zeta_N$ such that $K\subseteq ...
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1
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134
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Compact subgroups of a linear group over non-Archimedean local field
$\DeclareMathOperator\GL{GL}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers.
Is it true that any compact subgroup of $\GL_n(\mathbb{F})$ is conjugated to ...
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1
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How to get a ball in the nonvanishing locus of a polynomial in $\mathbb Z_p[x_1,\cdots,x_n]$ canonically?
Suppose $f\in \mathbb Z_p[x_1,\cdots,x_n]$, and consider $D(f):=\{(𝑥_1,…,𝑥_𝑛)∈ℤ^𝑛_𝑝:𝑓(𝑥_1,…,𝑥_𝑛)≠0\}\subset \mathbb Z_p^n$. How to calculate a radius $r$ from the datum of $f$ such that $D(f)$...
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