Tagged Questions

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.

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Deducing Skolem-Mahler-Lech theorem from a $p$-adic interpolation result.

Let $p$ be a prime number, let $d$ be an integer greater than $1$ and let $f\in\mathbb{Z}_p[X_1,\cdots,X_d]^d$. I have already proved the following $p$-adic interpolation theorem: Theorem 1. (...
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A number $a$ is a square in $\mathbf{Q}$ if and only if it is a square in $\mathbf{R}$ and $\mathbf{Q}_p$ for all primes $p$

Problem from Schikhof's Ultrametric Calculus. As I understand it, the intersection of $\mathbf{R}$ and all $\mathbf{Q}_p$ is just $\mathbf{Q},$ so it seems that $x^2-a$ having a zero in $\mathbf{Q}$ ...
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Limit of a sequence in $\mathbb{Z}_p$ (J.-P. Serre, p-adic equations)

In a proof of a theorem in chapter 2 "p-adic equations" in "A Course in Arithmetic" from J-P Serre there is one conclusion that I don't understand. Here is the theorem I'm talking about (excluding the ...
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There are several ways to construct the real numbers: completion of the rationals w.r.t. the euclidean metric, dedekind completion of the rationals, infinite continued fractions, etc. Each of these ...
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Characters with values on the $p$-adic complex field $\mathbb{C}_p$?

Characters $\psi : G \to \mathbb{C}$ from abelian groups $G$ to the complex field $\mathbb{C}$ are well-known and appear all over. Is there an analogue for the $p$-adic complex numbers $\mathbb{C}_p$, ...
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Subgroups of finite index of the split maximal torus of $GL_n(\mathbb{Z}_p)$.

Let $E_{i,j}$ be the $n \times n$ elementary matrices. Let $G=GL_n(\mathbb{Z}_p)$. Let $T_G$ be the split maximal torus of $GL_n(\mathbb{Z}_p)$. Let $\Theta$ be the subgroup of $T_G$ consisting of ...
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If $a \in \mathbb{Z}_5$ and $a \equiv \pm1 \text{ }(\text{mod }5)$, does there exist $x \in \mathbb{Z}_5$ where $x^2 = a$?

If $a \in \mathbb{Z}_5$ and $a \equiv \pm1 \text{ }(\text{mod }5)$, does there exist $x \in \mathbb{Z}_5$ where $x^2 = a$? I know we want to use Hensel's Lemma somehow to assess this question, but I'm ...