Questions tagged [p-adic-numbers]
The p-adic number system for any prime number p extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems
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Is the p-adic density of the image of a polynomial always rational?
This question was previously posted here on MSE.
Let $P(x)$ be a polynomial with integer coefficients, and let $p$ be a prime number. For $n\in\mathbb N$, let $I_n$ be the number of integers $i\in\{1,\...
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2-adic Logarithm and Resistance of n-dimensional Cube
Resistance across opposite vertices of n-dimensional cube with each edge at one ohm resistance is
$$R_n=\sum_{k=0}^{n-1}\frac1{(n-k){n\choose k}}=\frac1n\sum_{k=0}^{n-1}\frac1{{n-1\choose k}}.$$
The ...
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If a unitsquare is partitioned into 101 triangles, is the area of one at least 1%?
Update: The answer to the title question is no, as pointed out by Tapio and Willie. I would be more interested in lower bounds.
Monsky's famous theorem with amazingly tricky proof says that if we ...
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Automorphisms of $\mathbb C_p$
I am looking for a non-trivial automorphism $\sigma$ of $\mathbb C_p$ such that $\sigma(\mathbb Q_p)\subset\mathbb Q_p$.
If $\mathbb C_p$ were spherically complete, then by Hahn-Banach theorem, that ...
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Does the equation $1 + 2 + 3 + \dots = -\frac{1}{12}$ have a natural $p$-adic interpretation?
Consider the equation
$$1 + 2 + 3 + 4 + \cdots = - \frac{1}{12},$$
"proved" by Ramanujan Euler. One correct way to interpret this is that $\zeta(-1) = - \frac{1}{12},$ where $\zeta(s) = \sum_{n = 1}^{\...
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A list of proofs of the Hasse–Minkowski theorem
I am currently doing a project in which I intend to include the most insightful possible proof of the Hasse–Minkowski theorem (also known as the Hasse principle for quadratic forms, among other names) ...
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What is the $p$-adic Langlands conjecture for $\mathbf{GL}_1$?
In the Boston conference on Fermat's Last Theorem (Summer 1995), Barry Mazur said (around 15m into the video) about class field theory that
If you are a number-theorist and you want to cheer ...
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A quantitative version of Hensel's Lemma
I've been reading some papers on Igusa zeta functions, and they seem to be implicitly using a "quantitative version" of Hensel's Lemma, which also asserts the number of lifts of a $\mathbb{Z}/p\mathbb{...
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is there a p-adic implicit function theorem?
I am trying to find a good reference for a version of the implicit function theorem over $p$-adic manifolds. None of the texts I have consulted ( including "$p$-adic numbers, $p$-adic analysis, and ...
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What's the number of solutions of the quadratic equation $x_1^2+\dots+x_m^2=0$ over finite ring $\mathbb{Z}/p^n$?
I want to calculate the number of solutions to the quadratic equation $$x_1^2+\dots+x_m^2=0$$ where $m$ is odd (a given number) and $x_i\in\mathbb{Z}/p^n$ for a given prime number $p$ and a given ...
user178596
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Do algebraic completion/topological completion of fields always terminate? If so, are they unique?
Take the field $\mathbb{Q}$, If we complete it topologically with respect to the Euclidean norm, we get $\mathbb{R}$, then if we complete it algebraically, we get $\mathbb{C}$.
On the other hand, the ...
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Non-existence of "higher" Artin map
So rank $1$ local Langlands is special in as that it is given by the Artin map
$$\text{GL}_1(K)\to G_K^{ab},$$
whereas in the higher rank (to the best of my knowledge) there doesn't exist a map
$$\...
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Integral Tate-Sen theory
Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $C=\widehat{\overline{K}}$ be the completion of the algebraic closure of $K$. Let $\mathscr{O}_C$ be the ring of integers in $C$, and let $G_K$ ...
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Polynomial reducible modulo every integer
Hi,
let $f\in\mathbb{Z}[X]$ be a monic polynomial. Assume that the reduction of $f$ modulo $m $ is reducible for all integers $m\geq 2$.
Q1: Is $f$ reducible in $\mathbb{Z}[X]$ ?
I've thought ...
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Choice of digits for extensions of $\mathbb{Q}_p$
I am interested in writing (in base $p$) elements of the maximal unramified extension $\mathbb{Q}_p^{\mathrm{unr}}$ of $\mathbb{Q}_p$, or (its completion) the field $\mathrm{W}(\mathbb{F}_p^{\mathrm{...
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Computing preimage of element under norm map of quadratic extension of $2$-adic fields
Let $F$ be a $2$-adic field, i.e. a finite extension of the $2$-adic numbers $\mathbb{Q}_2$. Suppose that I have a quadratic extension $E = F(\sqrt{d})$ of $F$. Given a unit $\alpha \in \mathcal{O}_F^\...
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Is there a classification of the $p$-adic normed division algebras?
A normed division algebra over $\mathbb{R}$ is a pair $(A,\lVert{-}\rVert)$ with
$A$ an $\mathbb{R}$-algebra with a unit $1_A$;
$\lVert{-}\rVert\colon A\to\mathbb{R}_{\geq0}$ a norm on $A$;
such ...
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The stabiliser group of an isotropic quadratic form over $\mathbb{Q}_p$ is non-compact?
Let $\mathbb{Q}_p$ denote the $p$-adic integers. Let $V$ be a $\mathbb{Q}_p$-vector space and $Q : V \rightarrow \mathbb{Q}_p$ be a non-degenerate integral quadratic form. We say that the pair $(Q,V)$ ...
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Does $17x^4+y^2=-1$ have solution in $\Bbb{Q}_2(\sqrt{-5})$?
This question raised when I tried to calculate $2$-Selmer group of elliptic curve $E:y^2=x^3+17x$ over $\Bbb{Q}(\sqrt{-5})$.
$17x^4+y^2=-1$ does not have solution in $\Bbb{Q}_2$
(https://math....
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