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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.

2321 questions
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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 the rational numbers in the same way the real…
celtschk
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5 answers

Why are $p$-adic numbers and $p$-adic integers only defined for $p$ prime?

It makes perfect sense to speak of a base $10$ digit expansion. Why does it not make sense to speak of $10$-adic numbers or $10$-adic integers?
Mario
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20
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p-adic numbers vs real numbers

Could anyone give a concrete example of a p-adic number that is not a "real number"? that is, do we create "new numbers" (non real numbers) by completing Q with a non Archimedean norm? If so, what are they?
Bourama Toni
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12
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Square roots in the $p$-adics

Suppose I want to know whether $\sqrt{7}\in\mathbb{Q}_5$, or more generally, whether $\sqrt{n}\in\mathbb{Q}_p$ for $n\in\mathbb{Z}$, $p$ an odd prime. What are the techniques for determining this? Am I supposed to "lift" it from…
Nick
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11
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5 answers

How do I differentiate between $\sqrt{5}$ and $-\sqrt{5}$ in the $11$-adics - and should I even try to?

I've been playing around with the $11$-adic numbers lately, and in particular, the values of $\sqrt{5}$. I have two different approximations of this value: $a_1=\ ...937785904$A$44_{11}$ $a_2=\ ...1733251$A$6067_{11}$ Of course, as how in the real…
AKemats
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9
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2 answers

Easy way to determine the primes for which $3$ is a cube in $\mathbb{Q}_p$?

This is a qual problem from Princeton's website and I'm wondering if there's an easy way to solve it: For which $p$ is $3$ a cube root in $\mathbb{Q}_p$? The case $p=3$ for which $X^3-3$ is not separable modulo $p$ can easily be ruled out by…
pki
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What's $-\frac{1}{27}$ in the p-adic ring $\Bbb Z_2$?

What's $-\frac{1}{27}$ in $\Bbb Z_2$? I was naively thinking take the repeating string of the standard binary representation of $3^{-n}$, put it to the left of the point and you get $-3^{-n}$ in $\Bbb Z_2$. Hey presto, it works for $-\frac13$ and…
it's a hire car baby
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8
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1 answer

Constructing the complex p-adic numbers

I'm reading through "$p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions" by Koblitz to learn about p-adic numbers. In chapter 3, he describes the construction of $\Omega$ (a.k.a. $\Omega_p$), the completion of the algebraic closure…
Zachary
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7
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1 answer

Ultrametric inequality

I am having trouble seeing the following consequence of the ultrametric inequality, which is supposed to be immediate. If $|x+y|\leq \max{\{|x|,|y|\}}$, then, equality holds when $|x|\neq |y|$. I looked up three books/notes and all of them just say…
Milo Kunis
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7
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What is a group of type $(2,2)$?

This is a statement in Serre's A Course in Arithmetic (p. 18). If $p\ne2$, the group $\mathbb{Q}_p^*/\mathbb{Q}_p^{*2}$ is a group of type $(2,2)$. What is a group of type $(2,2)$?
abc
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What is $e, \pi, \ln 2,...$ etc in p adic?

What is $e, \pi, \ln 2,...$ etc in p adic? And how to flip digits of decimal points? Does p-adic have their own constants? 10 adic base.
waqar ahmad
  • 109
5
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1 answer

Intersection of p-adic integers

If we have two primes, $p$ and $p'$ and the $p,p'$-adic integers $\mathbb{Z}_p$ and $\mathbb{Z}_{p'}$. Then the integers inject into both of the rings, so we can say that $\mathbb{Z} \subset \mathbb{Z}_p \cap \mathbb{Z}_{p'}$. My question is, are…
user106209
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5
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$\mathbb{Q}^*$ closed in the finite ideles?

I want to consider the (topological) group of 'finite' ideles: If $$\mathbb{A}_\text{fin} = \widehat{\prod}^{\mathbf{Z}_p}_p \mathbf{Q}_p$$ (the 'hat' indicates the so-called restricted product topology) then $$\mathbb{I}_\text{fin} :=…
Fabian Werner
  • 919
5
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1 answer

About two 10-adic integer $x$ satisfies $x^2 = x$.

Consider about the 10-adic integer $x = \dots1787109376$ satisfies $x^2 = x$ and the 10-adic integer $y = \dots8212890625$ satisfying $y^2 = y$. Define $a(n)$ as the $n$th digit of $x$: $$a(0) = 6, \quad a(1) = 7, \quad a(2)= 3, \quad \dots…
TOM
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5
votes
3 answers

square root in 2-adics

$\newcommand\Q{\mathbb Q} \newcommand\Z{\mathbb Z}$I am a bit confused on the square root of 2-adics. I am pretty sure I am mixing some steps in an algorithm. To be precise, I am trying to solve an exercise in Koblitz' book p-adic numbers, p-adic…
quantum
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