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.

1994 questions
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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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19
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Divergent series and $p$-adics

If we naïvely apply the formula $$\sum_0^\infty a^i = {1\over 1-a}$$ when $a=2$, we get the silly-seeming claim that $1+2+4+\ldots = -1$. But in the 2-adic integers, this formula is correct. Surely this is not a coincidence? What is the connection…
19
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Is the algebraic closure of a $p$-adic field complete

Let $K$ be a finite extension of $\mathbf{Q}_p$, i.e., a $p$-adic field. (Is this standard terminology?) Why is (or why isn't) an algebraic closure $\overline{K}$ complete? Maybe this holds more generally: Let $K$ be a complete Hausdorff discrete…
17
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Proving $2+2^2/2+2^3/3+2^4/4+\cdots=0$ elementarily

In the first chapter of Gouvea's intro to $p$-adics, there's a heuristic argument that $$ \frac{2}{1}+\frac{2^2}{2}+\frac{2^3}{3}+\frac{2^4}{4}+\cdots=0 \tag{$\ast$}$$ as $2$-adic numbers, since it's the Mercator series for $\ln(-1)$ and…
17
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2 answers

Various $p$-adic integrals

Here are three (possibly different) definitions of $p$-adic integrals that I have encountered during my self-studies. First of all, here is what Vladimirov, Volovich and Zelenov write at the beginning of their book on mathematical physics: As the…
16
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5 answers

Classical number theoretic applications of the $p$-adic numbers

I am sure we can all agree that the $p$-adic numbers are highly fascinating objects in their own right - just as the closely related theory of valuations. Having independently read up on the $p$-adic numbers for a few weeks now, I have so far only…
16
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6 answers

$3^n$ does not divide $4^n+5$ for $n\geq 2$

Question as in the title : does anyone know how to prove that $3^n$ does not divide $4^n+5$ for $n\geq 2$ or find a counterexample ? My thoughts : (1) I have checked that this is true for $n\leq 1000$. (2) I asked a similar question recently, and it…
Ewan Delanoy
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16
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Are all nonarchimedean valuations discrete?

I am studying valuation theory on the way to local class field theory, and the texts I have looked at immediately focus on discrete valuations in developing the theory of nonarchimedean valuations. Why? Are there nondiscrete nonarchimedean…
Vitaly Lorman
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16
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Why is $O_K\otimes \mathbb{Z}_p\cong \oplus_{\mathfrak{p}|p}O_{K,\mathfrak{p}}$?

In my old number theory notebook this is stated as a fact. However, I ran into problems when I tried to prove it. First let me state the (supposed) theorem accurately: Theorem (?) Let $K$ be a number field with ring of integers $O_K$. Let $p$ be a…
15
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Roadmap to Iwasawa Theory

I haven’t found any posts on this, so I figured I would ask. Beyond learning basic algebra (rings, groups, fields) and complex analysis, what must one study if they want to start learning a good amount of iwasawa theory? In what sequence should they…
15
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1 answer

What are the branches of the $p$-adic zeta function?

I'm reading the book $p$-adic Numbers, $p$-adic Analysis, and Zeta-Functions by Neal Koblitz. In it, Koblitz wants to iterpolate the Riemann Zeta function for the values $\zeta_p(1-k)$ with $k \in \mathbb{N}$, where $\zeta_p(s) = (1-p^s)\zeta(s)$.…
14
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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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14
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Is there an explicit embedding from the various fields of p-adic numbers $\mathbb{Q}_p$ into $\mathbb{C}$?

For any field of p-adic numbers $\mathbb{Q}_p$, one can construct the field $\mathbb{C}_p$, the metric completion of one of its algebraic completions. By the axiom of choice, we can prove this to be isomorphic to the usual field of complex numbers…
14
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What do the $p$-adic roots of unity look like?

I know that $\mathbb{Z}_p$ has all the $p-1^{st}$ roots of unity (and only those). Is it true that mod $p$ they are all different? Meaning, is the natural map $\mathbb{Z}_p \rightarrow \mathbb{F}_p$, restricted to just the roots of unity, bijective?
Nicole
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14
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On 5-adic representation of square root of -1

Let $\alpha \in \mathbb{Z}_5$ be the solution to $f(x):=x^2+1=0$ such that $\alpha \equiv 2 \, (\text{mod} \,5)$ (that we obtain by Hensel's lemma). Then $$ \alpha = \sum_{k=0}^{\infty} a_k \, 5^k$$ where $a_0=2$ and $0\leq a_{n+1}<5$ is the only…