# 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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### Completion of an extension $K|\mathbb{Q}$ w.r.t. a non archimedean abs. value, isomorphic to $K\cdot \mathbb{Q}_p$

As the title suggests I want to prove the following result: given $\mathfrak{p}|(p)$, prime ideal of $\mathcal{O}_K$ over $p$, we have the canonical absolute value induced by it on $K$, and we can ...
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### $p$-adic digits via character sums

Let $p$ be a prime and let $n = \sum_{k=0}^\infty n_k p^k$ be a $p$-adic integer with each $0 \leq n_k \leq p-1$. Fix $0 \leq c \leq p-1$. Is there a way to check whether the $i$-th digit $n_i$ equals ...
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### Finding maximum number of factors in n!

I am quite new to $v_p()$ problems, and would like to know if anyone prove that $v_n(n!)\le n/2$? Basically, what I mean is that prove that for all positive integers $n$, the amount of factors of $n$ ...
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### If $B$ is an abelian group, then is $B{\otimes}_{\mathbb Z}{\mathbb Z}_p$ isomorphic to ${\varprojlim}B/p^{n}B$?

I could get the easy map from $B{\otimes}_{\mathbb Z}{\mathbb Z}_p$ to ${\varprojlim}B/p^{n}B$ but I could not find the map in the opposite direction. Please help me. Thank you!!
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### Finding an example of a non-rational p-adic number

We know that every rational number can be written as a $p$-adic integer with expansion $\sum\limits_{n=-m}^\infty a_n p^n$, where $a_n\in\{0,\dots,p-1\}$ and $m\in\mathbb{N}$; therefore there exists ...
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### Prove that the canonical p-adic expansion of $a\in\mathbb{Q}_p$ terminates

I am having trouble getting started with the following problem: Prove that the canonical p-adic expansion of $a\in\mathbb{Q}_p$ terminates (so $a_i = 0$ for all $i \geq N$) if and only if a is a ...
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### 7-adic expansion of a rational number

I know that every rational number has a unique 7-adic expansion, now I need help proving that $7/36=\sum\limits_{i=0}^nn7^n$ as a 7-adic integer. I tried using properties of this fraction, like adding ...
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### How to calculate $\log_p x$ in p-adic analysis?

I'm studying p-adic analysis and recently I've learned about the p-adic logarithm function but I can't understand very well how the process of calculating the value should be done. As an exercise I'm ...
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### Fast way to check if finite extension is unramified?

Consider the $5$-adic quadratic extension $\mathbb{Q}_{5}(\sqrt{5})/\mathbb{Q}_{5}$. I want to check if this extension is unramified, where unramified means that the corresponding extension of ...
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### Solution of a equation in $\Bbb{Z}_p$

Let $m \in \Bbb{Z}_p$ be fixed. Let $a_1,...a_l$ be fixed integers. I am trying to find out solutions of the equation $m=x_1^{a_1}...x_l^{a_l}$ where $x_1,...,x_l\in \Bbb{Z}_p$. Here $x_1,...x_l$ are ...
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In Alain Robert's A Course in p-adic Analysis, the author uses Hensel's Lemma to analyze quadratic extensions of $\mathbb{Q}_p$. He wants to calculate the index of $(\mathbb{Q}^*_p)^2$ in $\mathbb{Q}^*... 0answers 20 views ### Prove that the sequence$(p^n)^{∞}_{n=1}$for p a prime is a null sequence with respect to$| · |_p$. Prove that the sequence$(p^n)^{∞}_{n=1}$for p a prime is a null sequence with respect to$| · |_p$. Here is what I have: a null sequence is a sequence that maps to zero.$| · |_p=p^{-vp(\cdot)}$... 1answer 111 views ### A theorem on p-adic power series This is a question about a proof of a theorem in many books on$p$-adic numbers and I don't seem to understand one of the directions. The theorem is Let$f(X) = \sum\limits_{n=0}^\infty a_nX^n \in \...
Koblitz states in his book on p-adic numbers on page 84: Suppose that $\alpha \in \mathbb Q$ is such that $1 + \alpha$ is the square of a nonzero rational number $a/b$. Let $S$ be the set of all ...