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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### Inertia field example of $\mathbb{Q}_5(\sqrt[4]{50})$

Let $L = \mathbb{Q}_5(\sqrt[4]{50})$ and denote by $E$ the inertia field of the extension $L / \mathbb{Q}_5$. Write down a prime element $\pi_E$ of $\mathcal{O}_E$ with $L = E(\sqrt{\pi_E})$. Can ...
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### What is this notion of continuity? Pt. 2

Let $f : \mathbb{Z}_p\to \mathbb{Z}_q$, where $\mathbb{Z}_p$ ($\mathbb{Z}_q$) are the $p$-adics ($q$-adics) for $p\neq q$. I have encountered the class of all $f$ satisfying \tag{1}\...
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### What is this notion of continuity?

Let $f : \mathbb{Z}_p\to \mathbb{R}$, where $\mathbb{Z}_p$ are the $p$-adics. I have encountered the class of all $f$ satisfying $$\forall x\in\mathbb{Z}_p, f(x) = \lim_{n\to\infty}f(x\bmod p^n)$$ ...
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### conductors of representations coming from jacobians of curves

Let $C$ be a curve defined over $\mathbb{Q}$, and we denote by $J:=Jac(C)$ its Jacobian. For a prime $l$, we define by $V_l(J)=T_l(J)\otimes \mathbb{Q}_l$. There is a natural action of the absolute ...
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### Calculation of Hilbert symbol over $\Bbb Q_p$ where $p$ is an odd prime

Let $a,b\in \Bbb Q$ be rationals and $p$ an odd prime. I want to calculate the Hilbert symbol $(a,b)_p$. According to https://en.wikipedia.org/wiki/Hilbert_symbol#Hilbert_symbols_over_the_rationals, ...
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### If $χ_p$ is ramified then $χ(p) = 0$

Consider suppose that $χ_{idelic}$ is the idelic lift of the Dirichlet character $χ$. In my text, it says that if the local character $χ_p$ is ramified that is there exists some u $\in Z_p^{x}$ such ...
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### Which of the equations has solutions in $\mathbb{Z}$?

Consider the three equations of whom two only have trivial solutions in $\mathbb{Z}$. Determine a non-trivial solution of the third. $3x^2+5y^2=7z^2$ $5x^2+7y^2=3z^2$ $3x^2+7y^2=5z^2$ My Approach: I ...
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### Determine for the numbers $r \in \{-1,2,6,\frac{4}{5}\}$ the set of primenumbers $p$, such that r is a suqare in the field $\mathbb{Q}_p$.

Determine for the numbers $r \in \{-1,2,6,\frac{4}{5}\}$ the set of primenumbers $p$, such that r is a suqare in the field $\mathbb{Q}_p$. I want to use the following Theorem for that. Theorem: Let $p$...
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### The equation $x^3=b$ has a solution in $\mathbb{Q}_p$ if and only if $3|n$ and $x^3=u$ has a solution in $\mathbb{Z}_p^{\times}$.

Let $p$ be prime. Suppose $b=p^nu \in \mathbb{Q}_p^{\times}$, where $u \in \mathbb{Z}_p^{\times}$. The equation $x^3=b$ has a solution in $\mathbb{Q}_p$ if and only if $3|n$ and $x^3=u$ has a solution ...
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### The equation $x^3=u$ has a solution in $\mathbb{Z}_p^{\times}$ if and only if $x^3 \equiv u \mod p$ has a solution.

Let $p \neq 3, p \in \mathbb{P}$ and $u \in \mathbb{Z}_p^{\times}$ . The equation $x^3=u$ has a solution in $\mathbb{Z}_p^{\times}$ if and only if $x^3 \equiv u \mod p$ has a solution. My Approach: ...
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### Does there exists an integer-coefficient polynomial that extracts the highest digit of an integer in base p? [closed]

Given an odd prime $p$, a positive integer $1 \lt n$, and an integer $x \in \mathbb{Z}/p^n\mathbb{Z}$, does there exist an an integer-coefficient polynomial that extracts the highest digit of $x$? The ...
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### Calculate the $p$-adic absolute value of $\frac{1}{p^7-p^5+p^3}$

I want to solve the following exercise: Calculate the $p$-adic absolute value of $\frac{1}{p^7-p^5+p^3}$. My Approach: For the $p$-adic absolute value, we have the following rules: $|xy|_p=|x|_p |y|_p$...
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### Tamely ramified extensions of $K_{\mathfrak p}^{\mathrm{unr}}$

I have some doubts regarding this statement, which I don't know if it's true: Statement: Let $K_{\mathfrak p}$ be the completion of a number field w.r.t. the $\mathfrak p$-adic valuation, and let $K_0$...
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### When a quotient of $\mathbb Z_p[[x,y]]$ is a regular local ring

Consider the power series in two variables with p-adic coefficients. Namely consider $\mathbb Z_p[[x,y]]$. Moreover let $c\in\mathbb Z_p$ and construct the quotient: $$W=\mathbb Z_p[[x,y]]/(xy-c)$$ ...
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### Similar matrices with entries in $p$-adic numbers

It is known that Given a rational matrix $Q$, if the characteristic equation of $Q$ has integral coefficients, then $Q$  is similar ( over $\mathbb Q$ ) to a matrix with integral entries. Do we ...
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### Prove that $v_{p}(K^{\times}) = \frac{1}{e}\Bbb{Z}$, where $e \mid n = [K:\Bbb{Q}_{p}]$

I start learning about Algebraic Number Theory. To see the definition of $p$-adic valuation see this. I was trying to prove the result: Prove that $v_{p}(K^{\times}) = \frac{1}{e}\Bbb{Z}$, where $e$ ...
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### Exactness of the sequence $0\rightarrow\mathbb{Z}_p\xrightarrow{p^n}\mathbb{Z}_p\xrightarrow{\varepsilon_n} A_n\rightarrow 0$

On page 12 of the book A Course in Arithmetic by Serre, the author proves that the sequence $0\rightarrow\mathbb{Z}_p\xrightarrow{p^n}\mathbb{Z}_p\xrightarrow{\varepsilon_n} A_n\rightarrow 0$ is exact,...
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### Understanding the (integer)$p$-adic numbers and its canonical embedding

I did learn the following: The integer p-adic numbers are defined as $\mathbb{Z}_p:=\{(\overline{x_k}) \in \prod_{k=0}^{\infty} \mathbb{Z}$ / $p^{k+1}\mathbb{Z}:x_{k+1} \equiv x_k \mod p^{k+1}\}$ for ...
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### Comparing the norm map of ramified quadratic extensions of $\mathbb Q_p$ and of $\breve{\mathbb Q}_p$

Let $p$ be an odd prime number. Let $E$ be a ramified quadratic extension of $\mathbb Q_p$ with non-trivial Galois involution denoted by $a \mapsto \overline{a}$. Let us fix a uniformizer $\pi \in E$ ...
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### Difference between $PGL(2,\mathbb{Q}_p)$ and $GL(2,\mathbb{Q}_p)$ in terms of determinant

I'm trying to understand the difference between $PGL(2,\mathbb{Q}_p)$ and $GL(2,\mathbb{Q}_p)$ in terms of determinant. For example, in the real case, $PGL(2,\mathbb{R})$ is $GL(2,\mathbb{R})$ up to ...
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### Norm $\text{N}_{L/ \Bbb Q_p}(\pi_L)$ of Uniformizer of finite extension of $p$-adics

Let $K/ \Bbb Q_p$ a finite extension of $p$-adic field $\Bbb Q_p$ of degree $[K:\Bbb Q_p]=n$ and let $\pi_L$ be a uniformizer of $L$. Question: Can we say something interesting / "distinguished&...
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### Upper estimation for Multiplicative Conductor of Extension of $\Bbb Q_p$

Let $L/\Bbb Q_p$ be a finite extension of $p$-adics of degree $n$. Let $f$ be the minimal number such $1 +(p)^f:=U^f \subset \text{Norm}_{L/ \Bbb Q_p}(L^*)$. The ideal $(p)^f$ is also called ...
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### Can $p$-adic functional analysis be applied to algebraic geometry?

I learned that $L^2$ estimate based on funtional analysis could be used to prove some vanishing theorems of cohomology of coherent analytic sheaves over complex manifolds or complex projective ...
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### Is $\sqrt{3}$ in the field of 7-adic numbers $\mathbb{Q}_7$?

I am trying to solve the following question: Show that $\sqrt{3} \notin \mathbb{Q}_7$, where $\mathbb{Q}_p$ means the field of $p$-adic numbers. My approach was based on examining the quadratic ...
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### Some questions about almost étale extension of local field

I'm in trouble understandig the example and proof of Example 4.2.2 in Denis Benois's lecture note about $p$-Adic Hodge Theory, he gives the following definition: A finite extension $E/F$ of non-...
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### $\mathfrak{p}$-adic valuation of a norm

Given extension of Number fields L/K and prime ideal $\mathfrak{P} \in O_L$ lying over $\mathfrak{p} \in O_K$. Let $\hat{L}$ and $\hat{K}$ be the completions of L and K respectively. And let e and f ...
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