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p-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of p-adic numbers.

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### Does maximally incompleteness cause nonvanishing of the extension of maximal ideal of a valuation ring by rank 1 free module?

In B. Bhatt's lecture notes, Remark 4.2.5 says ... $\operatorname{Ext}_R^2(k,R)$ is non-zero if $K$ is not spherically complete. which amounts to the following pure algebraic question. Statement ...
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### Is equation $y^3+x y + x^4 + 4 = 0$ solvable locally (in ${\mathbb Q}_p$ for all $p$)?

When finding out whether an equation in 2 variables has rational solutions (or, equivalently, whether an algebraic curve has any rational points), many authors recommend checking the local solubility ...
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### Bernoulli distributions and $p$-adic measure on $K$

The $p$-adic field $\mathbb{Q}_p$ has topological basis of open sets of the form $a+p^N \mathbb{Z}_p$ for $0 \leq a \leq p^N-1$ and $N \in \mathbb{Z}$. These are indeed compact open sets. One can ...
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### A p adic limit of a binomial coefficient

Let $0 \leq a \leq p^n$ be a number coprime to p. Consider the following sequence of binomial coefficients: $$B_k = \binom{p^{n+k}}{p^ka}$$ as $k\to \infty$. If I did the computation right, the p-...
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### Local global principle for a system of polynomial equations

Suppose $T$ be a system of polynomials homogenous of degree 2 solvable over $\mathbb{R}$ and $\mathbb{Q}_p$ for all primes $p$. So, can we claim that $T$ is solvable over $\mathbb{Q}$? I think as of ...
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### Can perfect numbers be seen $p$-adically?

It is well known that all even perfect numbers are of the form $N=(2^{q}-1).2^{q-1}$ with $M_{q}:=2^{q}-1$ a Mersenne prime. As the very defining property of such a perfect number is to fulfill the ...
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### Computing the ring of power-bounded elements in an affinoid algebra

Let $K$ be a finite extension of $\mathbb{Q}_p$, and let $A$ be an affinoid $K$-algebra, i.e. $A$ is isomorphic to a quotient of the Tate algebra $K\left<T_1,\dotsc,T_n\right>$ for some $n$. ...
Let $a$ be algebraic over $K:=\mathbb F_q\left(\!\left(\frac1T\right)\!\right)$. Can one extend continuously the hyperderivatives on $K(a)$? Recal that the hyperderivative $D_h$ over $K$ is defined by ...