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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Power series in $\mathbb{Q}_5$

Could you help me to find the first five positions of the power series in $\mathbb{Q}_5$ of $\frac{1}{2}$? How can I do this? Is there a general formula?
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The additive $p-$ adic valuation of $\mathbb{Q}_p$: $$w_p: \left\{\begin{matrix} \mathbb{Q}_p \rightarrow \mathbb{Z} \cup \{\infty\}\\ p^m u \mapsto m\\ 0 \mapsto \infty \end{matrix}\right.$$ $$\... 1answer 91 views How can we show that it is an integer 5-adic number? Show that the number \frac{3}{8} is an integer 5-adic and calculate the first five positions of its power series in \mathbb{Q}_5. Could you explain me how we can conclude that \frac{3}{8} is ... 1answer 172 views Ring of the integer p-adic numbers \mathbb{Z}_p Let the ring of the integer p-adic numbers \mathbb{Z}_p. Could you explain me the following sentences? It is a principal ideal domain.$$$$The function \epsilon_p: \mathbb{Z} \to \mathbb{Z}_p... 1answer 199 views Computing a uniformizer in a totally ramified extension of \mathbb{Q}_p. Do you know how to compute a uniformizer of \mathbb{Q}_p(\zeta_{p^n},p^\frac{1}{p})? Where \zeta_{p^n} is a primitive p^n-th root of 1 and p is an odd prime. 1answer 55 views Tate's thesis - continuous map from a local field to circle group I am currently reading Decomposition of Unitary Representations defined by a discrete subgroups of nilpotent groups, by C.C. Moore. It is metioned that if \mathbb{K} is a p-adic field in his ... 0answers 71 views Does a generalization of the Teichmuller-character for non-prime arguments exist? Rereading an older article on Fermat-quotients in which I'd applied some p-adic-rationale I find now, that my method for the representation of bases b which allow high fermat-quotients b_{p,m}^{p-1}... 1answer 112 views Prove some properties of the p-adic norm I need to prove that the p-adic norm is an absolute value in the rational numbers, by an absolute value in a field K I mean a function |\cdot|:K \to \mathbb{R}_{\ge 0} such that:$$\begin{...
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Given two numbers $a,b \in \mathbb{Z}$, how do we prove that the $p$-adic number of $\gcd(a,b)$ is the same as the minimum for the $p$-adic number of $a$ and the $p$-adic number of $b$? Does this ...
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Why $\mathbb{Q}_p^{ur} \neq \widehat{\mathbb{Q}_p^{ur}}$?

Why is $\mathbb{Q}_p^{ur}$ not complete? And is there a criterion to know when $K^{ur} = \widehat{K^{ur}}$ ? (where $K$ is a p-adic field, i.e. a field of characteristic 0 that is complete with ...
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P-adic Lie groups - Representation theory

I am quite familiar with the Representation Theory for locally compact groups and nilpotent Lie groups. I want to start with the study of $p$-adic Lie groups representation theory, in particular ...
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Showing that Q is not complete with respect to the p-adic absolute value

I am looking at some notes that give an example of a Cauchy sequence that doesn't converge in $\mathbb{Q}$ with respect to the $p$-adic absolute value. Their example is to let $1 < a< p-1$ and ...
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Show that i is an element of the p-adic integers if and only if p congruent to 1 mod 4

This exercise was given in a graduate course on Local Class Field Theory. We want to prove that $i\in \mathbb{Z}_p$ (the $p-$adic integers) if and only if $p\equiv 1 \mod 4$. For $\Rightarrow$, we ...
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What is $\hat{\mathbb{Z}}$?

I have been reading a bit about unramified extensions. If $K$ is $\mathbb{Q}$ or $\mathbb{Q}_p$ (p-adics), then there is a maximal unramified extension $K^{nr}$ of $K$. Then I have read in some notes ...
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Strong Approximation for adelic quaternionic groups

Let $H$ be a definite quaternion algebra defined over $\mathbb{Q}$, unramified at $p$, and ramified at $\infty$. Denote by $D$ the multiplicative group $H^\times$ divided by its center. In this paper,...
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Let $E/\mathbb{Q}$ be an elliptic curve. For a prime $p$, we have the mod $p$ representation \begin{equation*} \bar{\rho}_{E,p}: G_{\bar{\mathbb{Q}}/\mathbb{Q}} \rightarrow Aut(E[p]) \end{equation*} ...
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Suppose that we are working over a nonarchimedean local field $F$, for instance $\mathbb{Q}_p$. Which semisimple algebraic groups (or Lie groups) over $F$ are simply-connected? In particular, I am ...
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The space $C_c^\infty(\mathbb{Q}_p^*)$ of smooth compactly supported functions on $\mathbb{Q}_p^*$

Let $p$ be prime. Let $\mathbb{Q}_p^*$ be the multiplicative group of the field of $p$-adic numbers. We call a function $f:\mathbb{Q}_p^*\rightarrow\mathbb{C}$ smooth if it is invariant under ...
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Let $p$ be prime such that $p\equiv 2\bmod 3$. Show that for every $a\in \mathbb Z,p\nmid a$ there is a $x\in \mathbb Z_p$, where $\mathbb Z_p$ is the field of the p-adic integers, such that $x^3=a$.
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Torsion of finitely generated $\mathbb Z_ \ell$-module finite?

Let $M$ be a finitely generated $\mathbb Z_ \ell$-module, where $\ell$ is a prime number and $\mathbb Z_\ell$ is the ring of $\ell$-adic integers. Let $T$ be its torsion submodule. Is $T$ finite? In ...
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$p$-adic expansion

I have just touched on this topic, please guide me along. If I have a prime number $p=10^{10}+19$, and a $p$-adic number $\alpha=\frac{16}{17}$. How do I derive its $p$-adic expansion? Thanks in ...
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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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Is there a non-compact open subset of the ring of $p$-adic intergers $\Bbb{Z}_p, p$ a prime?

Can anyone give me some idea? I cant find it at all.
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Proof of Hasse's principle for quadratic equations

I am currently tackling the following problem. Problem Consider the equation $x^2 = q,$ where $q \in \mathbb{Q}$. Show this has a rational solution $x$ in $\mathbb{Q}$ if and only if there are ...
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Proving the p-adic numbers $\mathbb{Q}_p$ form a field

I am trying to prove that $\mathbb{Q}_p$ forms a field. However, I am unsure of the best way to go about proving it. If I work with the power series representation of p-adic numbers I run in to ...
Direct proof of compactness of $\mathbb{Z}_p$
Let $\mathbb{Z}_{p}$ be completion of $\mathbb{Z}$ with respect to $p-$norms. Actually I know that $\mathbb{Z}_{p}$ is bijective to Cantor set, which is compact, therefore by homeomorphism, it is also ...