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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### Why $E_1(\mathbb{Q}_p)\cong\mathbb{Z}_p$

I read an article where it is said: $E_1(\mathbb{Q}_p)\cong \mathbb{Z}_p$ where $E$ is an elliptic curve over $\mathbb{Q}_p$ and $E_1(\mathbb{Q}_p)=\{P\in E(\mathbb{Q}_p):\tilde{P}=\tilde{O}\}$. The ...
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### Where is the flaw in this argument with $p$-adic extensions?

I cannot find what I am missing in the following computation. Let $K=\mathbb{Q}_p(p^{1/{(p-1)p^{\infty}}})$ and $L=\mathbb{Q}_p(\zeta_{p^{\infty}})$, where $\zeta_{p^n}$ is a primitive $p^n$-th root ...
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### intuition for lattices in p-adic (or other non-Archimedean) vector spaces?

I could use some help to jumpstart my intuition for lattices in vector spaces over non-Archimedean fields, like $\mathbb{Q}_p$ and $\mathbb{F}_q((t))$. I have some intuition for $\mathbb{Z}$-lattices ...
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### Analytic continuation of $f(x)=\sqrt{\frac{1-x}{1-x^p}}$ over the p-adics

Consider the power series $f(x)=\sqrt{\frac{1-x}{1-x^p}}$ over the algebraic closure of $\mathbb{Q}_p$, defined by $f(0)=1$. What can be said about an analytic continuation "in the form of Mittag-...
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### valuation of a derivative in a completion

Let $q$ be a power of a prime $p$ and $w$ be an irreducible polynomial of $\mathbb F_q[T]$. Denote by $\mathbb C_w$ the completion of an algebraic closure of $K_w$, the completion of $\mathbb F_q(T)$ ...
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### p-adic expansion of roots of unity [closed]

Let $w$ be an n-th root of unity, I have two questions 1) What are the conditions on the prime $p$ such that $w\in \mathbb{Z}_p$, and if it is the case what is the p-adic expansion of an n-th root of ...
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### If two monic polynomials of $\mathbb{Z}_p[X]$ (p-adic integer coefficient) are relatively prime modulo p, then do they generate $\mathbb{Z}_p[X]$?

I'm currently reading a paper of Rene Schoof, and I got stuck in a line. And I'm trying to check the above sentence. Although that seems to be elementary, I hope someone can give me a counterexample ...
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### Endomorphisms of the p-adic group $(\mathbb Z_p,+)$

Does there exist an endomorphism of $(\mathbb Z_p,+)$ of finite order different of $x\mapsto\xi x$ where $\xi$ is a root of unity in $\mathbb Z_p$? Thanks in advance
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### Mori: p-adic and real hemispheres of the mathematical universe?

I recall having read, some time ago, a beautiful and poetic opening of an article (or was it a book?). From memory, it was by Shigefumi Mori, and talked about the (mathematical) universe consisting of ...
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### Inverse of reduction mod $p$ functor?

I have a very general, and possibly not very precisely stated question, which comes up quite often in my work, and I would be very happy to be able to address. To my dismay, I only have some very ...
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### Nick Katz observation: “the rationality of the zeta function!”

In the proceedings "Algebraic Geometry - Arcata 1974" edited by R. Hartshorne there is an article by Nick Katz called "$p$-adic $L$-functions via moduli of elliptic curves". He starts by recalling $p$-...
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### Zero digits of a p-adic algebraic number

This question might be too simple and I just don't see something very obvious, so I apologize in advance if that is so. Let $p$ be a prime number and let $\mathbb Z_p$ denote the ring of $p$-adic ...
For each prime number $p$ and number field $k$, there exists at least one extension $k_{\infty}/k$ with Galois group isomorphic to $\mathbb{Z}_p$, the cyclotomic $\mathbb{Z}_p$-extension. If $k_p/k$ ...
Let $p$ be an odd prime, and let $E/\mathbb{Q}_p$ be a finite extension that contains a primitive $p$-th root of unity $\zeta_p$ but not a primitve $p^2$-th root of unity $\zeta_{p^2}$. Let \$a,b \in E^...