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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### Find the invers of $4 \in \mathbb{Z}_5$ (The 5-adic integers)

I am trying to solve this question, however I don´t seem to have the correct expression of the inverse to solve the remaining part: QUESTION: Find the inverse of 4 in $\mathbb{Z}5$. Use your answer to ...
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### Infinitely large Galois extensions of $\mathbb Q$ inside $\mathbb Q_p$

Let $p$ be a fixed prime number and denote $\mathbb Q_p$ the field of $p$-adic numbers. For each positive integer $n$, I would like to construct a finite Galois extension $K/\mathbb Q$ of degree at ...
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### Two seemingly different totally ramified extension,$\Bbb{Q}_p(ζ_{p^n})$ and $\Bbb{Q}_p(p^{1/n})$

$\Bbb{Q}_p(ζ_{p^n})$ and $\Bbb{Q}_p(p^{1/n})$ are both totally ramified extension over $\Bbb{Q}_p$ each has extension degree $p^n-p^{n-1}$ and $n$. The former can be regarded as Lubin Tate extension,...
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### How to prove that in p-adic rationals every sphere is open?

I‘m a little bit confused with this proof question, because it does intuitively not make any sense: Let r = pm for m in $\mathbb{Z}$ and p prime, let q be a rational number and |.|p be the p-adic norm ...
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### What would be the compositum $K \cdot \mathbb{Q}_p(\sqrt u)$?

Consider $K$ be an unramified extension of $p$-adic field $\mathbb{Q}_p$ of degree $n$. I want to compute the compositum $K \cdot \mathbb{Q}_p(\sqrt u)$, where $u^2=-1$. Since $K$ is unramified ...
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### Intersection of $\Bbb{R}$ and $\Bbb{Q}_p$ [duplicate]

Intersection of $\Bbb{R}$ and $\Bbb{Q}_p$ For distinct prime $p$ and $q$,intersection of $\Bbb{Q}_p$ and $\Bbb{Q}_q$ is just $\Bbb{Q}$, but what about $\Bbb{Q}_p$ and $\Bbb{R}$ ? $\Bbb{R}$ is ...
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### On the $p$-adic expansion of an integer in $\mathbb{Z}_p$.

I am currently reading upon $p$-adic integers, and I have a quick question related to $p$-adic expansions in $\mathbb{Z}_p$. Given a $p$-adic expansion in $\mathbb{Z}_p$, is there a criterion to ...
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### Why $p$-adic logarithm is continuous on $\mathbb{Q}_p^\times$?

Neukirch defined, in Algebraic Number Theory, Neukirch (5.4) p. 136, the $p$-adic logarithm on $1 + p\mathbb{Z}_p$ as $\log(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots$ Then, he extends this ...
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### Does $\Bbb{Q}_2$ has $\sqrt{-1}$?

Does $\Bbb{Q}_2$ has $\sqrt{-1}$? I tried to use Hensel lemma as usual. Let $f(x)＝x^2＋1$. But if some $a∈\Bbb{Z}$, $f(a)＝0$, then $f'(a)$ can always divide by $2$. So I cannot use Hensel lemma. Could ...
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### Is $1＋p/2!＋p^2/3!＋p^3/4!・・・$ convergent in $\Bbb{Z}_p$?

Let $p$ be an odd prime.Is $1＋p/2!＋p^2/3!＋p^3/4!・・・$ convergent in $\Bbb{Z}_p$? I know $1＋p＋p^2/2!＋p^3/3!＋・・・$ converges but what about the titled case ?
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### Is always $e' \mid e$ true?

Let $K$ be a finite extension of $\mathbb{Q}_p$ with ramification index $e$. Then we have $v(\pi)=\frac{1}{e}$, where $\pi$ is the uniformizer in the ring of integers $O_K$ of $K$. Let $b \in K$ be an ...
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