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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Show that this fractal contains every odd, positive integer

Let $f(x)=2x+\frac{1}{3}$ Let $g(x)=\frac{2x-1}{3}$ $f^n$ represents composition of $f$ Let $G_0=\{1\}$ Let $F_{m}=\{f^n(x):x\in G_{m}, n\in\mathbb{N_{\geq0}}\}$ Let $G_{m+1}=\{g(x):x\in F_m\}$ Show that for any given odd, positive integer $p$…
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Interpretation of open balls with the p- adic metric

I am going to give a presentation about p-adic numbers. While studying what p-adic numbers even are, I got stuck at the geometric interpretation of the open balls. I have read many times that every point in those balls are considered to be the…
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What is the measure of any given section of this variant of the Sierpinski carpet? Relationship between p-adics and $\pi$.

The Sierpinski carpet is a $1\times 1$ square with the centre $9^{th}$ removed, then the centre $9^{th}$ of the eight remaining disjoint squares removed, and so on $n$ times, with $n\to\infty$, as described here: Wolfram If you take a section…
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Is the topology of the p-adic valuation to the unramfied extension complete?

Consider $\mathbb Q_p^{\text{ur}}$ the maximal unramified extension of the p-adic numbers. Suppose that on $\mathbb Q_p$ we have the usual absolute value that extends $|\frac{a}{b}|_p=\frac{1}{p^{v_p(a)-v_p(b)}}$ to $\mathbb Q_p$. Now it is known…
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Purpose of Robert exercise on p-adic Numbers

I've been working through Alain Robert's A Course in $p$-adic Analysis and there's an exercise regarding $K((x))$ for a field $K$ in the first chapter that I'm hoping is more related to the topic than the obvious "It's another thing with an…
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How to prove Tate module of the Tate Curve is not crystalline

Let $K/\mathbb{Q}_p$ be a finite extension and let $q \in K^{\times}$ be such that $|q| <1$. Let $E_q:= \bar{K}^{\times}/q^{\mathbb{Z}}$ be the Tate curve where $q^{\mathbb{Z}}:= \{q^n| n\in \mathbb{Z}\}$. If we define $V_p(E_q):=…
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How does p-adic metric notation work?

I want an expression that returns the highest power of $5$ that divides $x-1$. i.e. i want to return an element of $\{1,5, 25, 125,\ldots\}$ Is $\lvert x-1 \rvert_5$ correct notation for the 5-adic metric of $x-1$? And does that mean that if…
samerivertwice
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$p$-adic perfect square condition.

Let $a$ be an integer $p$-adic number for $p\ge2$. Prove that $a$ is a perfect square iff $\exists$ such integer $x$ that $a \equiv x^2 mod$ $p$. It is easy to prove $\Rightarrow$: if $a$ is a perfect square then $\exists$ such $b = (...,b_2, b_1,…
Gleb Chili
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$a\in\mathbb{Z}_p^{\times}\iff \forall n\geq 0, \gcd(n,p(p-1))=1,\,\exists\, b_n\in\mathbb{Q}_p \text{ with } a=b_n^n$

If $p\in\mathbb{Z}$ is prime, prove that: $1)$ If $a\in\mathbb{Q}_p^{\times}$, then $a\in\mathbb{Z}_p^{\times}\iff \exists\, b_n\in\mathbb{Q}_p^{\times}$ such that $a=b_n^n$ for all $n>0$ with $\gcd(n, p(p-1))=1$ $2)$…
rmdmc89
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algebraic closure of $\mathbb{Q}_p$ is not complete

A paper I'm reading says that the algebraic closure of $\mathbb{Q}_p$ is not complete, by using for example the Baire theorem. Wikipedia says the Baire theorem says that a complete metric space is a Baire space (meaning a countable intersection of…
usr0192
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Show that all triangles in $\mathbb{Z_p}$ are isosceles

I need to show that all triangles in $\mathbb{Z_p}$ are isosceles. Here, $$\mathbb Z_p = \lim_{\longleftarrow} A_n \space, (A_n = \mathbb{Z}/p^n \mathbb{Z})$$ Now, the topology on $\mathbb{Z_p}$ can be defined as $$d(x,y) = e^{-v_p(x-y)}$$ where…
Dark_Knight
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Why is $O_{\mathbb{C}_p}/p$ not perfect?

I was reading the beautiful Brinon and Conrad's introduction to $p$-adic Hodge theory (link) and I came to the introduction of de rham period ring. During the book, they always mention that the quotient of the ring of integers of the completion of…
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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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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 group topology) if $G$ has a base of $e$ formed…
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Primitive roots of unity in p adic integers

Question is to find a condition when can be $\mathbb{Z}_p$ has an $m$th root. First non trivial and simple root of unity is fourth root of unity $m=4$. So, we want to know when does $f(x)=x^2+1$ has solution in $\mathbb{Z}_p$. Suppose it has then it…
user87543
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