# Tagged Questions

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### Non-standard extensions of $p$-adic fields

Does there exist a non-standard extension of a non-Archimedean field (such as the construction $*\mathbb{R}$ out of $\mathbb{R}$ or the surreals $\mathbb{S}_\mathbb{R}$, not to mention their ...
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### History of $p$-adic numbers

I'm interested in learning about the historical motivation and development of $p$-adic numbers. I haven't been able to find any books on the topic. I'd appreciate any references, including to more ...
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### $\mathbb{Z}_p[\text{PGL}_2(\mathbb{F}_{p^n})]$-modules

Let $p$ be prime. Let $\mathbb{Z}_p$ be the $p$-adic integers. I'm intersted in $\mathbb{Z}_p[\text{PGL}_2(\mathbb{F}_{p^n})]$-modules of low rank over $\mathbb{Z}_p$ (rank $2$ is already very ...
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### Open problems involving p-adic numbers

I am in my final year of my undergraduate degree, and I'm doing a project on p-adic numbers, and in particular, trying to find Galois groups of simple extensions of $\mathbb{Q}_p$ (this is a Galois ...
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### Characterization of integers which has a $2$-adic square root

Does anyone know an "elementary" proof of the following theorem? Let $k \neq 0$ be a rational integer. Then $k$ admits a square root in $\mathbb{Z}_2$ if $k = 4^a (8b+1)$ for some $a \in \mathbb{N}$, ...
I am interested to know if there is a standard name for the inverse limit, $\hat{\mathbb{Z}}_!$, say, of the inverse system of rings $$\ldots \rightarrow \mathbb{Z}/((n+1)!)\mathbb{Z} \rightarrow ... 0answers 41 views ### presentation of the inertia group of p-adic fields It is known (see here) that the absolute Galois group of a p-adic number field K (ie a finite extension of \mathbb Q_p) is topologically finitely presented for any odd prime p. Is it also ... 1answer 204 views ### Roots of unity in \mathbb Q_p Is there a way to get the number of solutions of an equation like x^n=1 in the p-adic field \mathbb Q_p, where p is a prime and n a positive integer? I know that for n=p-1 there are p-1 ... 1answer 110 views ### Choosing an isomorphism \tau:\overline{\mathbf{Q}}_p\simeq\mathbf{C}; how do things depend on choice of \tau? I sometimes see arguments that begin by choosing an isomorphism of fields \tau:\overline{\mathbf{Q}}_p\simeq\mathbf{C}, and then defining some property in terms of this isomorphism. I'm not so ... 4answers 331 views ### Divergent series and p-adics If we naïvely apply the formula$$\sum_0^\infty a^i = {1\over 1-a} when $a=2$, we get the silly-seeming claim that $1+2+4+\ldots = -1$. But in the 2-adic integers, this formula is correct. Surely ...
As we all know, the ring $Z_p$ can be constructed as the projective limit of the rings $Z/p^{n}Z$. Now is there any generalization such as the p-adic completions of a Dedekind Domain? This might be ...