Questions tagged [p-adic-number-theory]
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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Connection between roots in $Q_p$ and $Z_p$
I'm working on some problems where I have to find solutions in $Z_p$ and $Q_p$ of polynomials of the form $ax^2+by^2=1$. I've seen Hensel's lemma for solution over $Z_p$. For solutions over $Q_p$. I'...
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How to prove the p-adic expansion of negative p-adic integers ends with p-1?
I just started calculating with p-adic numbers and one of the questions I'm stuck on is how I can prove that for a negative integer the p-adic expansion ends with an infinite amount of digits p-1. I ...
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On the concept of primary element
Let $\ell$ be an odd prime number,$\zeta_{\ell}:=e^{2\pi i/\ell}$, $F:=\mathbb{Q}(\zeta_{\ell})$ be a cyclotomic field, $\mathcal{O}_F$ be its integer ring and $\lambda:=1-\zeta_{\ell}$.
[Ireland-...
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What is the decomposition of global units $1+\mathfrak{p}$?
Let $p \geq 2$ be prime and $K=\mathbb Q(\zeta_p),~\zeta^{p}=1$ with ring of integers $\mathcal{O}_K$. we denote by $\mathfrak{p} \mid p$ the prime ideal of $K$ dividing $p$. Let $K_{\mathfrak{p}}$ be ...
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Why's $v_L (x)=\frac{1}{n}v_K(N_{L/K}(x))$ integer for unramified local field extension $L/K$?
If $K$ is a local field and $L/K$ is a finite extension, then the valuation $v_K$ can be extended uniquely to a valuation $v_L$ of $L$ such that $v_L$ restricted to $K$ is equal to $v_K$. This is one ...
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Does $2y^2=4+17x^4$ have solutions in $\Bbb{Q}_2(\sqrt{-3})$?
Does $2y^2=4+17x^4$ have solutions in $\Bbb{Q}_2(\sqrt{-3})$ ?
My try is the following, is this correct ?
Look at $2$ adic valuation $v$ of $\Bbb{Q}_2(\sqrt{-3})$ on both side.
$v(2y^2)=1+2v(y)$,
$v(4+...
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Question about the multiplicative inverse of a p-adic number
I am reading the text of Gouvea about p-adic numbers and I am trying to show that every $\alpha\in \mathbb{Q}_{p}$ has a multiplicative inverse. In the case where $$\alpha =\sum_{i\geq 0}a_{i}p^{i}, \ ...
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Does $\sqrt{-1}y^2=-1+17x^4$ have solution in $\Bbb{Q}_{17}(\sqrt{-1})$?
Does $\sqrt{-1}y^2=-1+17x^4$ have solution in $\Bbb{Q}_{17}(\sqrt{-1})=\Bbb{Q}_{17}$?
For $y=0$, $x=1/17^{1/4}$ does not belong to $\Bbb{Q}_{17}$.
For $x=0$, $y=(\sqrt{-1})^{1/2}$ does not belong to $\...
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Why is the generalized definition of valued field automatically non-Archimedean?
The classical definition of a field $K$ with an absolute value $|\cdot|:K\to \mathbb{R}_{\geq 0}$ is that $\forall x,y\in K$
$x=0\Leftrightarrow|x|=0$
$|xy|=|x|\cdot|y|$
$|x+y|\leq |x|+|y|$
If the ...
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Variety structure of the Prüfer group
The group $\mathbb{R}/\mathbb{Z}$ rather famously is isomorphic to the algebraic group given by the variety $X^2 + Y^2 =1$ over $\mathbb{R}^2$ with group operation given by $(X_1,Y_1)\cdot (X_2,Y_2) = ...
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What is the quotient group $\mathfrak{q}^2/\mathfrak{p}^2\mathbb Z_p$?
Let $p \geq 2$ be prime and $K=\mathbb Q(\zeta_p),~\zeta^{p}=1$ with ring of integers $\mathcal{O}_K$. we denote by $\mathfrak{p} \mid p$ the prime ideal of $K$ dividing $p$. Let $K_{\mathfrak{p}}$ be ...
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Splitting on $p$ adic unit group
Let $p$ be a prime.
It is basic that following isomorphisms;
$\mathbb{Z}_p^{\times}/(1+p^n\mathbb{Z}_p )\cong (\mathbb{Z}_p/p^n\mathbb{Z}_p)^{\times} $,
$(1+p^n\mathbb{Z}_p)/(1+p^{n+1}\mathbb{Z}_p )\...
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Is there a relationship between the Leech lattice and the 2-adic expansion of certain numbers?
The 2-adic expansion of $1/1387$ is $\sum_{i=0}c_i2^i$ where
$$
c_i=\begin{cases}
[1,1][i] & i < 2, \newline
[0,0,0,0,1,0,1,1,1,1,1,1,1,1,1,1,0,1][(i-2)\mod18] & \text{otherwise}.
...
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9/10 as a 5-adic number
Working with p-adic integers is really simple, and usually if you want to do rational numbers, it's fairly easy; you just use long-division.
But here in the case 9/10 when we're working in the 5-adics;...
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Is the p-adic expansion of a rational number repating, and does every p-adic number with a repeating series correspond to a rational number? [duplicate]
Is the $p$-adic expansion of a rational number repeating after some point?
Does every $p$-adic number with a series that is repeating after some point correspond to a rational number?
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Why maximally incompleteness cause nonvanishing of the extension of maximal ideal of a valuation ring by rank 1 free module?
In B. Bhatt's lecture notes[1], Remark 4.2.5 says
... $\operatorname{Ext}_R^2(k,R)$ is non-zero if $K$ is not spherically complete.
which amounts to the following pure algebraic question.
Statement ...
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Does Taylor's theorem (Peano's form) generalize to other fields?
Let $\mathbb F$ be a field with a non-trivial absolute value, such as $\mathbb F_p(X)$ or $\mathbb Q$ or $\mathbb Q_p$, and let $f:\mathbb F\to\mathbb F$ be a function. Limits, continuity, and ...
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Is there a relationship between p-adic open ball and Infinite Ramsey Theorem?
The question I have is, a p-adic open ball could be considered a monochromatic subset in Zp?
From my understanding of p-adic open balls, they are a subset of Zp. But I'm not sure if they can be ...
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A generalization of Hensel's Lemma
I have problem with this exercise, I really don't know how to prove it.
Let $K$ be a field that is complete with respect to a non-Archimeadean norm $ \left| \cdot \right|$. We denote by $A = \{ x \in ...
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$\mathbb{Z}_p$ is a compact subgroup
Prove that $ \mathbb{Z}_p \subset \mathbb{Q}_p $ is a compact subgroup and show that this is the maximal compact subring.
Hint: use that every sequence of integers had a Cauchy sub-sequence with ...
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On the Iwasawa Algebra
I am reading Joaquin Rodrigues Jacinto's and Chris Williams' notes on $p$-adic $L$-functions http://www.ltcc.ac.uk/media/london-taught-course-centre/documents/Number-Theory---Full-Lecture-Notes-2017-...
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Classification of branched division maps on $\mathbb R/\mathbb Z$?
Consider the doubling map $x\mapsto 2x$ on $\mathbb R/\mathbb Z$. I'm interested in the collection of all right inverses of this map whose image is an interval. In other words, I want to know about &...
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Legendre symbol on $p$-adic integers $\mathbb{Z}_p$
I have seen that you can define the usual modular arithmetic on the $p$-adic integers:
For $a,b\in\mathbb{Z}_p$ and a prime $p$,
$$a\equiv b\pmod{p}\iff (a-b)/p\in \mathbb{Z}_p.$$
My question is, can ...
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$p’$ elements of general linear group over $\mathbf{Z}_p$
Let $p$ be a prime and let $\mathbf{Z}_p$ denote the ring of $p$-adic integers. Then there are infinitely many $p$-elements (i.e. those elements of order of a power of $p$) in ${\rm GL}_n(\mathbf{Z}...
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If there exists a prime $l$ such that $l$ spilts in $K$, why does that imply there exists a place of $K$, which satisfies $K_v \cong \Bbb{Q}_l$?
Let $K$ be a number field.
If there exists a prime $l$ such that $l$ spilts in $K$, why does that imply there exists a place of $K$, which satisfies $K_v \cong \Bbb{Q}_l$ ?
Here, $K_v$ denotes ...
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When is a norm of a formal power series over a local field a polynomial?
Let $K$ be a finite extension of $\mathbb Q_p$ and $L/K$ be a finite Galois extension. Then also $L(T)/K(T)$ and $L((T))/K((T))$ are Galois extensions with Galois group isomorphic to ${\rm Gal}(L/K)$ ...
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What is the smallest prime which satisfies $p \in {\Bbb{Q}_2(\sqrt{2} )^{\times}}^4$ and $p≡3,5$mod8?
What is the smallest prime which satisfies $p \in {\Bbb{Q}_2(\sqrt{2} )^{\times}}^4$and $p≡3,5$mod8?
The candidate for $p$ is $3,5,11,13,19,29,37,43,53,59,・・・$.
To consider which is $4$-th power in $\...
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Find prime number which satisfies $p \in {\Bbb{{Q}_2}^{\times}}^2-{{\Bbb{Q}_2}^{\times}}^4$
What is an minimal prime number $p$ which satisfies $p \in {\Bbb{{Q}_2}^{\times}}^2-{{\Bbb{Q}_2}^{\times}}^4$?
For $a \in \Bbb{Z}$, if $a≡1,3,4,7$mod8, then for each $a$, $\Bbb{Q}_2$ is $\Bbb{Q}_2(\...
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What does mean by "algebraic closure of $\mathbb Q$ in $\mathbb Q_p$"?
I want to understand the precise meaning of "algebraic closure of $\mathbb{Q}$ in $\mathbb{Q}_p$".
What does mean by "algebraic closure of $\mathbb Q$ in $\mathbb Q_p$" ? Isn't it ...
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What does the characteristic ideal of a f.g. torsion $\Lambda(G)$-module tell me about the arithmetic of the extension?
I am currently trying to learn Iwasawa theory and am following J. Coates and R. Sujatha's book 'Cyclotomic Fields and Zeta Values'.
The setup is the following:
Let $\mathcal{F}_n:=\mathbb{Q}(\mu_{p^{...
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What functions on 2-adic completion of the rationals are in the conjugacy class of $x+2^{\nu_2(x)}$?
Question
What functions on 2-adic space are in the conjugacy class of $f(x)=x+2^{\nu_2(x)}$?
A conjugacy class is the set of functions whose action conjugates to any other element of the class, in ...
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Let $K$ be a number field and $v$ be its place. Let $K_v$ be completion of $K$ at $v$. I want to prove $y^2=x^4-p$ has $K_v$ rational point.
Let $K$ be a number field and $v$ be its place. Let $p$ be a prime element of ring of integers of $K$. Let $K_v$ be completion of $K$ at $v$.
I want to prove $C: y^2=x^4-p$ has $K_v$ rational point.
I ...
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"p-adic completion" of rational function field
Let $q$ be a prime power, $\mathbb{F}_q$ the finite field of order $q$ and let $\mathbb{F}_q(T)$ be the field of rational functions over $\mathbb{F}_q$.
Similarly to $\mathbb{Q}$, the absolute values ...
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Mahler Series of the Gamma function
I saw the following identity few times while studying the subject of p-adic analysis, but I struggle a little bit trying to prove it. the identity is as follows
$\sum_{n=0}^\infty(-1)^{n+1}\frac{x^n}{...
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A question of parabolic subgroup of $SO(2n)$
Let $V, (,)$ be a $n$-dimensional quasi-split orthogonal space over a $p$-adic field $F$. Then there exist isotropic subspaces $X$ and $Y$ which are dual with respect to $(,)$. Let $\{x_1,…,x_{n-1}\}$ ...
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A uniformizer of a finite extension of $\mathbb{Q}_p$
Let $L$ be a finite extension of $\mathbb{Q}_p$, say $[L:\mathbb{Q}_p]=n$. The uniformizer of $\mathbb{Q}_p$ is $p$, let we assume that $\sqrt[n]{p} \notin L$ and consider $K=L(\sqrt[n]{p})$. Is $\...
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Showing an abelian pro-$p$ group is a $\mathbb{Z}_p$ module
I am rather new to $p$-adics, and I am trying to show that every abelian pro-$p$ group is a $\mathbb{Z}_p$ module. I know some partial answers already exist (About pro-$p$ groups), but I am having ...
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Torsion elements of general linear group over $\mathbf{Z}_2$
Let $p$ be a prime and let $\mathbf{Z}_p$ denote the ring of $p$-adic integers. Suppose that $n$ is an integer with $p>n+1$. Then it's well-known that the general linear group ${\rm GL}_n(\mathbf{Z}...
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Elements of a local field with trace in $\mathbb Z_p$
Let $K_p$ be a finite extension of $\mathbb Q_p$. Then we have the trace map:
$$
T:=\operatorname{Tr}_{K_p|\mathbb Q_p}:K_p\to\mathbb Q_p
$$
Is there any characterization of the open set $T^{-1}(\...
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What is the 2-adic expansion of 1/2?
I understand that -1/3 can be represented in 2-adics as $01010101...$, and 1/3 is just $1101010101...$. What about 1/2, though? Is it possible to represent it as an infinite expansion, or is the only ...
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Similarity of Lifting the Exponent Lemma in Pell numbers
Pell number is a term of the sequence $\{P_n\}$ determined by a recurrence relation
$$P_{n+2}=2P_{n+1}+P_n, P_0=0, P_1=1.$$
Let $v_p(x)$ be the $p$-adic valuation of an integer $x$ (the number of ...
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Strassmann's thoerem and irrationality measure of certain number
In this note from Keith Conrad, he explains an interesting application of Strassmann's theorem to the divergence of certain linear recurrence integer sequence. More precisely, the sequence defined as $...
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Is $\Bbb{Q}_p/\Bbb{Z}_p$ isomorphic to $\Bbb{Q}/\Bbb{Z}$ as aan abelian group?
Is $\Bbb{Q}_p/\Bbb{Z}_p$ isomorphic to $\Bbb{Q}/\Bbb{Z}$ as a group ?
My try: I know $\Bbb{Q}/\Bbb{Z}\cong \bigoplus_p:prime \Bbb{Q}_p/\Bbb{Z}_p$ as group.
If I could prove $\bigoplus_p:prime \Bbb{Q}...
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Radius of convergence in p-adics
Let $f(x)=\displaystyle{\sum_n a_nx^n}$ be a power series with coefficients in the field of p-adic rationals, $\mathbb{Q}_p$. Let $R_f$ be its radius of convergence and let $f'$ denote its term-by-...
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Show if $f:\mathbb{Z}_p\to\mathbb{Q}_p$ is continuous such that $f(n)=(-1)^n$, then $p=2$.
Let $\mathbb{Q}_p$ denote the p-adic rationals and same for the integers. Suppose $f:\mathbb{Z}_p\to\mathbb{Q}_p$ is continuous, with the additional property that at $n\in\mathbb{Z}^{\geq 0}$, $f(n)=(-...
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diophantine equations over $Q_p$
I need to show that there is no trivial solution to the equation $3x^2+2y^2-z^2=0$ in $Q_3$
So can I look at solutions in $F_3$ and if I didn't find any then assume that there is no solution in $Q_3$?
...
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What happens if I add an $\omega$th digit to the $p$-adic numbers?
If you add $3$-adic numbers like $\dots 111111 + \dots121212 = \dots 010100$ the digits all carry over, so it seems intuitively like you lose a digit at the $\omega$th place, as it's missing an $\...
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Take limit in p-adic integer ring: the difference between $\lim_{n \rightarrow \infty} \alpha^{n!}$ and $\lim_{n \rightarrow \infty} \alpha^{n}$
Let $p$ be a prime, $L / \mathbb{Q}_p$ be a finite field extension. Let $\mathcal{O}_L$ its valuation ring (ring of local integers) in $L$, $\mathfrak{m}_L$ be the unique maximal ideal, and the ...
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Is $\log_p(x^z)=z \log_p(x)$ for $z \in \mathbb Z_p$?
Let $\log_p$ denote the $p$-adic logarithm and $\mathbb Z_p$ be the ring of integers.
We know that $p$-adic logarithm has no base, but still we have $\log_p(x^n)=n \log_p(x)$ for $n \in \mathbb Z$ and ...
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$f(T)=T^2 + 1$ irreducibility over $p$-adic number polynomials, $\mathbb Q_p[T]$.
In relation to this previous post, Why is $x^2+1$ irreducible in $\mathbb{Q}_2[x]$., is there something we can say about $f(T) = T^2+1$ in generality over different choices of $p$ of $\mathbb Q_p[T]$? ...
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