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Questions tagged [p-adic-number-theory]

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 ...
Christian Schwacke's user avatar
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Maximal abelian extension of $K^{ur}$

Let $p$ be a prime integer and $K$ be a $p$-adic field. For each integer $n\geq 1$, let $K_n/K$ be the unique unramified extension of degree $n$. Denote by $K^{ab}_n$ the maximal abelian extension of $...
Lam's user avatar
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2-adic valuations of $k\cdot 3^n-1$

I was just playing around with numbers of the form $3^n-1$, and noticed that their 2-adic valuations have a nice, understandable pattern: $\left(v_2(3^n-1)\right)_{n\ge 1} = (1,3,1,4,1,3,1,5,1,3,1,4,1,...
G Tony Jacobs's user avatar
3 votes
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111 views

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 \begin{equation}\tag{1}\...
Mark Schultz-Wu's user avatar
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1 answer
178 views

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 $$ \forall x\in\mathbb{Z}_p, f(x) = \lim_{n\to\infty}f(x\bmod p^n) $$ ...
Mark Schultz-Wu's user avatar
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1 answer
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Understanding the proof of arithmetic local monodromy theorem

I am reading the proof of Grothendieck's $\ell$-adic local monodromy theorem in the book Theory of $p$-adic representations by Fontaine and and Ouyang. More concretely, let $K$ be a $p$-adic field, $...
Alexey Do's user avatar
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Valuation of $p-1-\zeta_{p^{n}}-...-\zeta_{p^{n}}^{p-1}$

Let $\zeta_{p^n}$ be a $p^n$th root of unity in $\overline{\mathbb{Q}}_p$. I want to compute the valuation of $$p-1-\zeta_{p^{n}}-...-\zeta_{p^{n}}^{p-1}$$ We can rewrite $p-1-\zeta_{p^{n}}-...-\zeta_{...
Desunkid's user avatar
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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 ...
did's user avatar
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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, ...
user302934's user avatar
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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 ...
Soumyadeep mandal's user avatar
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1 answer
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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 ...
NTc5's user avatar
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1 answer
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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 $p$...
NTc5's user avatar
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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 ...
NTc5's user avatar
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3 votes
1 answer
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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: ...
NTc5's user avatar
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3 answers
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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 ...
fofo's user avatar
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1 answer
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When is $x \in \mathbb{Q}_p$ a square?

Theorem: Let $p$ be odd prime. The map $\Phi_p{:\mathbb{Q}}_p^{\times}/\mathbb{Q}_p^{\times^{}2} \rightarrow \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$, $up^n \mapsto (\mu((\frac{u}{p})),n \...
NTc5's user avatar
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2 votes
1 answer
51 views

Lemma about $n$-adic numbers [closed]

In the book I am reading there is the following Lemma whose prove is "left for the reader": Lemma: For $n \in \mathbb{N},n\geq 2$ we define the ring of $n$-adic numbers $\mathbb{Z}_n=\{(\...
NTc5's user avatar
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1 answer
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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: $|xy|_p=|x|_p |y|_p$...
NTc5's user avatar
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4 votes
1 answer
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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 $K_0$...
Marta Sánchez Pavón's user avatar
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21 views

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: $$ W=\mathbb Z_p[[x,y]]/(xy-c) $$ ...
manifold's user avatar
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2 votes
1 answer
40 views

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 ...
ghc1997's user avatar
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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 $e$ ...
Afntu's user avatar
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2 votes
1 answer
76 views

How to determine if $x \in \mathbb{Q}_p$ is a square

I want to determine if a element $x \in \mathbb{Q}_p$ is a square. To do that I learned the following: Define $\mu:({-1,+1}, \cdot) \rightarrow (\mathbb{Z}/2\mathbb{Z},+), 1 \mapsto \overline{0}, -1\...
NTc5's user avatar
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4 votes
0 answers
65 views

$SO_3(\mathbb Q)$ contains a free group using 5-adic numbers

I am trying to show that $SO_3(\mathbb Q)$ contains a free group using 5-adic numbers, and more precisely using the matrices $M_1=\left(\begin{array}{ccc}1 &0&0\\0&\frac 35&-\frac{4}5\\...
Zumurud's user avatar
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1 answer
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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,...
Matheus Frota's user avatar
1 vote
2 answers
90 views

Forall $x \in \mathbb{Z}_p \setminus\{0\}$ can be written as $x= p^n u$ with $n \in \mathbb{N} \cup \{0\}$, $u \in \Bbb{Z}_p\setminus p \Bbb{Z}_p$

I want to understand the proof of the following statement: Let $\mathbb{Z}_p$ denote the $p$-adic integers. Every element $x \in \mathbb{Z}_p \setminus\{0\}$ can be written as $x= p^n u$ with $n \in \...
NTc5's user avatar
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2 votes
1 answer
94 views

Finding the $p$- adic representation of $\frac{1}{2}$ and $\frac{7}{4}$.

I want to solve the following Problems: Show that $\frac{1}{2}$ and $\frac{7}{4}$ are Elements in $\mathbb{Z}_5$ ($5$-adic integers) and calculate the first four coefficients of the power series ...
NTc5's user avatar
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0 votes
1 answer
71 views

Is it possible to show that $v_{p}(f(\alpha)) \in \frac{1}{r}\Bbb{Z}$, where $v_{p}(\alpha) \in \frac{1}{r}\Bbb{Z}$ and $f(\alpha) \neq 0$

Let $K/\Bbb{Q}_{p}$ be a finite extension. If $\alpha \in K \setminus \{0\}$. It is given that, $v_{p}(\alpha) \in \frac{1}{r}\Bbb{Z}$, where $r \neq 1$. Now is it true that, $v_{p}(f(\alpha)) \in \...
Afntu's user avatar
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0 votes
1 answer
44 views

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 ...
NTc5's user avatar
  • 609
1 vote
1 answer
45 views

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 $\pi \in E$ ...
Suzet's user avatar
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0 answers
33 views

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 ...
The way of life's user avatar
1 vote
0 answers
36 views

Examples of non-maximal orders in non-Archmidean fields that are not monogenic

Suppose I have a number field $K$ and a non-Archimedean place $v$. I know that the ring of integers $\mathcal{O}_{K_v}$ is monogenic. That means $\mathcal{O}_{K_v}=\mathbb{Z}_p[\alpha]$ for some $\...
Breakfastisready's user avatar
1 vote
0 answers
65 views

Why are these p-adic inequalities valid?

For rational $\alpha$ and rational $t\ne0$, I have the following recurrence relation of functions $f_i$ for natural numbers $i$. $$ t \cdot f_{n+1}=(\alpha+n) f_{n}-f_{n-1} $$ I know $f_0$ is 1 and $...
Sveti Ivan Rilski's user avatar
0 votes
1 answer
70 views

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&...
user267839's user avatar
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1 vote
0 answers
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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 ...
user267839's user avatar
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0 votes
1 answer
30 views

On the definition of an admissible representation of a $p$-adic group

Let $G$ be a reductive over a $p$-adic field $F$ and $(\pi,V)$ a (complex) smooth representation of $G(F)$. I know that the notion of admissible is defined as follows: for any compact open subgroup $...
youknowwho's user avatar
  • 1,479
6 votes
1 answer
99 views

Can the composite of $\mathbb Q_p$ and $\mathbb Q_\ell$ in $\mathbb C$ be $\mathbb C$?

I will motivate my question with two observations. Consider the standard embedding $\mathbb R\to\mathbb C$. Then, for any embedding $\mathbb Q_p\to \mathbb C$, we must have that the composite of $\...
Andrea B.'s user avatar
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0 answers
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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 ...
Siyuan Yin's user avatar
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0 answers
42 views

Computation of Norm

I am attempting an exercise from a Galois Theory problem sheet I found online. It asks: Let $p$ be an odd prime. Let $L_1 = \mathbb{Q}_p(\zeta_p) / \mathbb{Q}_p$ and let $L_2=\mathbb{Q}_p(\sqrt[p-1]{-...
Todd Burnett's user avatar
2 votes
1 answer
41 views

A limit problem in $p$-adic fields

I met a very concrete limit computation while dealing with some formulas in $p$-adic analysis. Let $F$ be a finite extension over $\mathbb{Q}_p$ with $p>2$ a prime. Let $E/F$ be a ramified ...
youknowwho's user avatar
  • 1,479
1 vote
0 answers
95 views

What are the ideals of the group ring $\Bbb Z_p[\Bbb Z/p\Bbb Z]$?

Let $p$ be an odd prime. Write $\mathbb{Z}_p$ to denote the ring of $p$-adic integers and set $G=\mathbb{Z}/p$. Is there a way to write explicitly what are all the ideals of the group ring $\mathbb{Z}...
debanjana's user avatar
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3 votes
2 answers
101 views

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 ...
Rafiz Sadique's user avatar
3 votes
3 answers
172 views

Taylor Series of $\frac{2p + 2p^2}{2+2p+p^2}$

I am trying to expand the following: $$ \frac{2p + 2p^2}{2+2p+p^2}. $$ Using the Taylor series at $a=0$, I get: $p - p^3/2 + p^4/2 - p^5/4 + \dots$. But in the book, it is also equated to $p + p^3 + ...
Naitik Mundra's user avatar
4 votes
1 answer
84 views

If $K/\mathbb{Q}_p$ totally ramified, is $\mathcal{O}_K / \pi^n \mathcal{O}_K \cong \mathbb{Z} / p^n \mathbb{Z}$?

As in the title, if $K/\mathbb{Q}_p$ is a finite and totally ramified extension, is $\mathcal{O}_K / \pi^n \mathcal{O}_K \cong \mathbb{Z} / p^n \mathbb{Z}$ for all $n \ge 1$? Obviously the case $n = 1$...
FlipTack's user avatar
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1 vote
1 answer
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How to check if $\mathbb{Q}_p(\alpha_n)/\mathbb{Q}_p$ is totally ramified?

Consider a polynomial $f(x) \in \mathbb Q_p[x]$ and consider its iterates $f^{\circ n}(x)$. Let $\alpha_n \in \bar{\mathbb Q}_p$ such that $f^{\circ n}(\alpha_n)=0$ for all $n \geq 1$. Suppose, I ...
MAS's user avatar
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2 votes
1 answer
71 views

Topological entropy of a Bernoulli Shift

I am approaching the world of entropy and I would like to have a few examples in mind. I know some chaotic systems that are topologically conjugated to the Bernoulli shift, so I would like to know the ...
Andrea Marino's user avatar
1 vote
0 answers
46 views

Understanding Robert's Proof of extending of the p-adic absolute value to field extensions

I am trying to understand a proof that the extension of the p-adic absolute value to a field extension $K$, given by $|x| = \sqrt[n] {|\text{N}_{K/\mathbb{Q}_p}(x)}|$ is an absolute value. Here $N_{K/...
ikey's user avatar
  • 133
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0 answers
45 views

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-...
Kevin's user avatar
  • 395
3 votes
0 answers
64 views

$\mathfrak{p}$-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 ...
Rick's user avatar
  • 391
1 vote
0 answers
56 views

disk of convergence of composition of p-adic functions

I watched a video that sketched out the regions of convergence for both the p-adic logarithm and the p-adic exponential functions. I thought about all this and asked myself: How would I find the ...
zeta space's user avatar

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