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Question 1

-3votes

0answers

60views

Understanding why applying a function on the congruence class 3 (mod 4) produces the ruler sequence [closed]

IntroductionI'm trying to understand how a specific function converts a set of specific odd numbers into the ruler sequence.I would like to understand why the following claim holds:$$ C(k) = \nu_2(...

DorkeDavid

1

asked14 hours ago

Question 2

0votes

0answers

29views

Could someone recommend me some references for algebraic groups? [closed]

I'm currently taking a number theory course, specifically a representation of reductive p-adic groups. To work through examples, could someone recommend me books or lecture notes that explain ...

metsu

1

askedyesterday

Question 3

2votes

0answers

108views

How to prove this inequality $2^n\le 2t+v_2\left( \sum_{i=t}^{2^{n-1}} \binom{2^n}{2i}\binom{i}{t}\right)$ $(1\le t\le 2^{n-1}$)?

Well, when I do the polynomial problem that my teacher gave me. I've tried new way to solve the problem by using p-adic $v_2$ but cannot solve it, here is the question:For two polynomials with ...

Tokisaki Kurumi

31

asked2 days ago

Question 4

0votes

0answers

22views

Intuition connecting the definition of norm of a number with its p-adic expansion

Let $\alpha\in K$ where $K$ is an algebraic extension of $\mathbb{Q}_p$ and $n:= [K : \mathbb{Q}_p]$.Let $f(x) : = x^n + a_{n-1}x^{n-1}+...a_0$ be a minimum polynomial of $\alpha$ over $\mathbb{Q}_p$....

emmy

256

asked2 days ago

Question 5

4votes

1answer

210views

A $3$-adic valuation regarding $ C_n = \frac{1}{3+\frac{2}{5+\frac{3}{7+\frac{4}{11+\frac{5}{13+\dots\frac{n}{p_n}}}}}} $ is always $1$?

My previously asked question motivated me to ask this question.For $n^{th}$ odd prime $p_n$,We define the following fraction:$$C_n = \frac{1}{3+\frac{2}{5+\frac{3}{7+\frac{4}{11+\frac{5}{13+\dots\...

Mr. Curious

321

askedNov 25 at 17:18

Question 6

1vote

0answers

102views

Splitting field of a polynomial over $\mathbb{Q}_p$?

How to find the degree and the ramification index of the splitting field of a certain polynomial over $\mathbb{Q}_p$? For instance, $f(x)=3+9x+3x^4+x^6$ over $\mathbb{Q}_3$?If $\gamma$ is one of the ...

8k14

311

askedNov 19 at 13:31

Question 7

0votes

1answer

53views

Is $f(x) \in K[x]$ is irreducible, if residual reduction $\bar f(x)$ has single linear factor?

Let $K/ \mathbb{Q}_p$ be a finite extension of the $p$-adic number field, with ring of integers $\mathcal{O}_K$ and maximal ideal $\mathfrak{m}$. Let $g(x) \in K[x]$ and let $\bar{g}(x)=(x-\alpha)^m$ ...

Learner

574

askedNov 17 at 19:08

Question 8

1vote

2answers

76views

The $p$-adic valuations of the pairwise differences for all distinct pairs of roots of an irreducible polynomial

Let $f$ be an irreducible polynomial over $\mathbb Q_p$ and $\alpha_1,\ldots,\alpha_n$ be its roots is an algebraic closure. It is known that the valuations of $\alpha_i$ are all equal. Is it true ...

8k14

311

askedNov 17 at 6:54

Question 9

1vote

0answers

132views

Is the reverse bit shift map times $3^n$ guaranteed to have sequences with no further multiples of $3$?

QuestionThe dynamical system:Let $f(x)=\begin{cases}(x+1)/2&&x\textrm{ odd}\\x/2&&x\textrm{ even}\end{cases}$is the the bit shift map with binary strings reversed. It terminates ...

Robert Frost

9,768

askedNov 16 at 17:51

Question 10

3votes

1answer

84views

Number of factors of a polynomial through Newton polygon argument

Let the Newton polygon of $f(x) \in \mathbb{Q}_p[x]$ has a single segment of horizontal length $L$ and slope $\lambda=\frac{h}{e},~\gcd(h,e)=1$.Is it true that $f(x)$ has $\frac{L}{e}$ many ...

Learner

574

askedNov 13 at 15:52

Question 11

1vote

0answers

71views

Factorization of a little higher degree polynomial over $p$-adic number field

Let $K=\mathbb{Q}_p(t)$ be the finite extension of the $p$-adic number field $\mathbb{Q}_p$, where $t=p^{1/13}$. With the help of Newton polygon argument, it seems the polynomial $$f(x)=(x^{p^2}-t^2)^...

Learner

574

askedNov 12 at 16:35

Question 12

8votes

2answers

473views

There are no integer solution to $a^p+ p b^p+ (10p+1) c^p=0$ for any $p$ prime

I am trying to prove that $(0,0,0)$ is the only integer solution of$$a^p+ p b^p+ (10p+1) c^p=0$$I found this question on an Italian forum for the particular case in which $p=11$. The question was ...

Marco

3,856

askedNov 11 at 17:08

Question 13

0votes

0answers

53views

Hensel's definition of equality for p-adic numbers.

I just started learning about p-adic numbers while reading Hensel's founding paper (I could not find the paper in english).Now he defines (if I got it right)The numbers $$A=a_0,a_1a_2\ldots,\quad A'...

edamondo

1,813

askedNov 6 at 11:55

Question 14

1vote

1answer

48views

$p$-adic completeness of convergent sequences in $\mathbb{Z}_p$

I feel really dumb asking this question since it feels really elementary, but I am having problems showing that the following $\mathbb{Z}_p$-submodules of $\prod_{\mathbb{N}} \mathbb{Z}_p$ are $p$-...

coconuthead

169

askedNov 6 at 1:20

Question 15

3votes

1answer

121views

If $f(x) \in \mathbb{F}_p[x]$ is irreducible , is the $p$-adic extension $\mathbb{Q}_p[x] / f(x)$ unramified?

I am reading Fernando Gouvea's p-adic numbers an introduction. However, I am not able to figure out the above question as most of the proofs appear as exercises. Can someone please cite a good ...

emmy

256

askedNov 3 at 12:11

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Questions tagged [p-adic-number-theory]

Understanding why applying a function on the congruence class 3 (mod 4) produces the ruler sequence [closed]

Could someone recommend me some references for algebraic groups? [closed]

How to prove this inequality $2^n\le 2t+v_2\left( \sum_{i=t}^{2^{n-1}} \binom{2^n}{2i}\binom{i}{t}\right)$ $(1\le t\le 2^{n-1}$)?

Intuition connecting the definition of norm of a number with its p-adic expansion

A $3$-adic valuation regarding $ C_n = \frac{1}{3+\frac{2}{5+\frac{3}{7+\frac{4}{11+\frac{5}{13+\dots\frac{n}{p_n}}}}}} $ is always $1$?

Splitting field of a polynomial over $\mathbb{Q}_p$?

Is $f(x) \in K[x]$ is irreducible, if residual reduction $\bar f(x)$ has single linear factor?

The $p$-adic valuations of the pairwise differences for all distinct pairs of roots of an irreducible polynomial

Is the reverse bit shift map times $3^n$ guaranteed to have sequences with no further multiples of $3$?

Number of factors of a polynomial through Newton polygon argument

Factorization of a little higher degree polynomial over $p$-adic number field

There are no integer solution to $a^p+ p b^p+ (10p+1) c^p=0$ for any $p$ prime

Hensel's definition of equality for p-adic numbers.

$p$-adic completeness of convergent sequences in $\mathbb{Z}_p$

If $f(x) \in \mathbb{F}_p[x]$ is irreducible , is the $p$-adic extension $\mathbb{Q}_p[x] / f(x)$ unramified?

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