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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### Can the p-adic be countable?

Recently arxiv submitted a new paper (Andrej Bauer, James E. Hanson, The Countable Reals) claiming an incredible theorem that Dedekind reals are not sequence-avoiding, and furthermore obtaining a ...
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### Bezout-type theorem for $p$-adic analytic plane curves

Let $p$ be a prime, and let $f,g \in \mathbb{Z}_p[[x,y]]$ be power series convergent on all of $\mathbb{Z}_p$. Suppose that the intersection of the analytic plane curves cut out by $f$ and $g$ is ...
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### Approximating $p$-adic power series by polynomials

Let $p$ be a prime, and let $f \in \mathbb{Z}_p[[x_1,\dots,x_d]]$ be a power series convergent on all of $\mathbb{Z}_p^d$. We make the following definition concerning the approximation of $f$ by ...
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### How do you prove that the series 5, 25, 625, ... can be continued forever to give a 10-adic solution to $n^2=n$? [closed]

How do you prove that the series 5, 25, 625, ... can be continued forever to give a 10-adic solution to $n^2=n$? Here's a proof for a different solution (...1787109376): https://oeis.org/A018248/...
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### Compact subgroups of a linear group over non-Archimedean local field

$\DeclareMathOperator\GL{GL}$Let $\mathbb{F}$ be a non-Archimedean local field. Let $\mathcal{O}$ be its ring of integers. Is it true that any compact subgroup of $\GL_n(\mathbb{F})$ is conjugated to ...
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### How to get a ball in the nonvanishing locus of a polynomial in $\mathbb Z_p[x_1,\cdots,x_n]$ canonically?

Suppose $f\in \mathbb Z_p[x_1,\cdots,x_n]$, and consider $D(f):=\{(𝑥_1,…,𝑥_𝑛)∈ℤ^𝑛_𝑝:𝑓(𝑥_1,…,𝑥_𝑛)≠0\}\subset \mathbb Z_p^n$. How to calculate a radius $r$ from the datum of $f$ such that $D(f)$...
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