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.

This tag is for $p$-adic number systems. Read more in this Wikipedia article.

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p-adic number for polynomial

I have three polynomials: $$f=x^5+x^4+4x^3+3x^2+3x$$ $$g=x^5+4x^3+2x^2+3x+6$$ $$p=x^2+3$$ Question: what is $v_p(fg)$? First, I have multiplied f and g: $h :=f \cdot g = x^{10}+x^9+8x^8+9x^7+24x^6+29x^5+36x^4+39x^3+27x^2+18x $ Then I have divided…
jublikon
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Quotient by squares of finite rank

Let $F$ be a number field and $\mathcal{O}$ its ring of integers. Why is $\mathcal{O}^\times/\mathcal{O}^{\times 2}$ of finite rank? In particular, what do we get for $K=\mathbf{Q}$? And for $K=\mathbf{Q}_p$?
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When does a polynomial with coefficients in the $p$-adic integers map all $p$-adic non-integers to $p$-adic non-integers?

Let $p\in\mathbb{N}_{\geq2}$ (not necessarily prime). How can one characterize all $P\in\mathbb{Z}_p[x]$ satisfying $P(\mathbb{Q}_p\setminus\mathbb{Z}_p)\cap\mathbb{Z}_p=\emptyset$ and all $P\in\mathbb{Z}_p[x]$ satisfying…
Mario
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counter example for Hasse Norm theorem

I was looking for examples for which Hasse-Norm theorem fails for abelian extensions. I found one which is given by $L=\mathbb{Q}(\sqrt{13},\sqrt{17})$ over $\mathbb{Q}$. The solution I saw shows $-1$ is a not global norm but is a local norm…
dragoboy
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What is the absolute value on $\mathbb{Z}_p[X]$ in this context?

Let $\mathbb{Z}_p$ denote the ring of p-adic integers for some prime $p \in \mathbb{Z}$. What is the absolute value on $\mathbb{Z}_p[X]$ ? The motivation for the question comes from a proof of Hensel's lemma where we construct a sequence of…
Improve
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Is $\mathrm{i}$ congruent $2$ or $3$ mod $5$?

The $5$-adic integers contain the $4$-th roots of unity $1$, $\mathrm{i}$, $-1$, and $-\mathrm{i}$. Solving the equations $x^2+1\equiv0\mod{5^k}$ gives two solutions: $\ldots3032431212$ and $\ldots1412013233$. Which of the two is $\mathrm{i}$ and…
Mario
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What follows from equality after reduction mod $p$ of closed subgroups of $\mathrm{GL}_n(\mathbb{Z}_p)$?

$\newcommand{\GL}{\mathrm{GL}}\newcommand{\dbZ}{\mathbb{Z}}\newcommand{\dbF}{\mathbb{F}}\newcommand{\dbN}{\mathbb{N}}$ I have a small question regarding closed subgroup of $\GL_n(\dbZ_p)$. Fix $n\in \dbN$ and let $p$ be a fixed prime. Let…
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does there exist a countably infinite normed field which is complete?

We know $Q$ with any of $p$ aidic norms is not complete for all primes including infinity. Now my question is can we make same conclusion for any countably infinite fields ? i.e does there exist a countably infinite normed linear field which is…
dragoboy
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If $H \subseteq G$ are finite abelian groups, how does $L^2(H)$ embed into $L^2(G)$?

If $H \subseteq G$ are abelian groups, how does $L^2(H)$ embed into $L^2(G)$? In particular, I am trying to find the Pontriyagin dual of the p-adic numbers. Example Here is a rather stunning visualization of the $2$-adic case. However, I could…
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Discrete $\mathbb{Z}_p[K]$ module with no K invariants is Trivial

I read the following fact in a paper: If $K$ is a finite $p$ group and $M$ a discrete $\mathbb{Z}_p[K]$ module such that $M^K=0$ then $M=0$. This seems obviously wrong to me, which leads me to believe I am not understanding the definitions…
rondo9
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p-adic expansions for powers of p in the denominator

I read this question and the answer to it, which deals witht the case when the number we are dealing with is a p-adic integer. How does one find a p-adic expansion for a number that has a p factor in the denominator? E.g. how do we find the 5-adic…
Nesa
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Find polynomials such that $uA+vB=R$

In my course in p-adic numbers I need to prove that any factorization mod $p$ in relatively prime factors lifst to $\mathbb{Z}_p$. One part of the question is the following: Given are two polynomials $A$ and $B$ in $\mathbb{F}_p[x]$ of degree $a$…
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Roots in different algebraic closure have the same multiplicative relations

Let $f \in \mathbb{Z}[x]$ be an irreducible monic polynomial, with $f(0) \ne 0$, and $p$ a prime number. Also, let $a_1, \ldots, a_n \in \overline{\mathbb{Q}}_p$ be all the roots of $f$ taken in the algebraic closure of the $p$-adic numbers, and let…
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Open balls that are also closed

Let $(X, d)$ be a topological space with ultrametric $d$. Under what assumption can we say that every open ball $B(a, r) = \{x \in X : d(a, x) < r\}$ is also a closed ball? This is certainly true for valued fields $X = K$ if their valuation group $G…
Santiago
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Prove that if $\| n \| \leq 1$ for all integers $n$, then $\| \cdot \|$ is a non-Archimedean norm.

I'm reading the book $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions by Neal Koblitz, and in Exercise 3 of Chapter 1 we are asked to show that if $\| \cdot \|$ is a norm on a field $F$ such that $\| n \| \leq 1$ for every integer $n$, then…
user279515
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