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.
1,982
questions
-1
votes
1
answer
98
views
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 ...
- 39
1
vote
1
answer
102
views
Prove that $f(x)=x^{3} + 2x^{2}+ 3x + 5$ has a root in the in $\mathbb{Z}_{13}$ (The 13-adics)
Prove that $f(x)=x^{3} + 2x^{2}+ 3x + 5$ has a root in $\mathbb{Z}_{13}$ (The 13-adics).
EDIT: NOTE: I HAD A TYPO WITH THE POLYNOMIAL'S COEFFICIENT!!!.
QUESTION: Prove that $f(x)=x^{3} + 3x^{2}+ 3x ...
- 39
-1
votes
0
answers
31
views
Show that: $\prod_p|x|_p = |x|^{-1}$
I am currently learning about p-adic numbers and I am struggling to solve the following problem:
Show that for any non-zero x $\in$ Q: $$\prod_p|x|_p = |x|^{-1}$$ where the product is over all primes ...
- 91
3
votes
1
answer
93
views
Find a polynomial of the form $F(x,y,z)$ of degree $3$ such that $F(a,b,c) = 0 \pmod{5}$ iff $a,b,c= 0 \pmod{5}$
I am trying to solve this question to study for my Number Theory final exam
QUESTION: Find a polynomial of the form $F(x,y,z)$ of degree 3 such that $F(a,b,c) \equiv 0 \pmod{5}$ iff $a,b,c \equiv 0\...
- 39
2
votes
1
answer
91
views
Why ${\rm Gal}\left (\overline{\mathbb{Q}_p}/ \mathbb{Q}_p\right )$ action on $ \overline{\mathbb{Q}_p}$ extends to its action on $\mathbb{C}_p$?
Why ${\rm Gal}\left (\overline{\mathbb{Q}_p}/\mathbb{Q}_p\right )$ action on $ \overline{\mathbb{Q}_p}$ extends to ${\rm Gal}\left (\overline{\mathbb{Q}_p}/ \mathbb{Q}_p\right )$ action on $\mathbb{C}...
- 390
0
votes
1
answer
37
views
why $|a|=|\sigma (a)|$, where $|\cdot|$ denotes absolute value on $\overline{ \Bbb Q_p}$?
Let $a\in\overline{ \Bbb Q_p}$, $\sigma\in Gal$( $\overline{ \Bbb Q_p}/ \Bbb{Q}_p$）, then, why $|a|=|\sigma (a)|$, where $|\cdot|$ denotes absolute value on $\overline{ \Bbb Q_p}$?
I think we should ...
- 390
0
votes
1
answer
41
views
+100
What is the local basis at $0$ of inverse limit topology?
What is the local basis of inverse limit topology at $0$?
For example, $\mathbb Z_p＝\lim\mathbb Z/p^n\mathbb Z$ has $$\{ \{(\cdots,α_0)\in\mathbb Z_p｜α_m＝・・・＝α_o＝0\} \mid m≧0\}$$as a local basis at $0$...
- 574
0
votes
0
answers
81
views
Intuition behind $\frac{a}{b} \equiv k \pmod{p} $
I am working with $p$-adic numbers at the moment and am having some trouble with a basic fact.
I know that for $\frac{a}{b}\in\mathbb{Q}$ there is a solution $k\in\mathbb{Z}$ to
$\frac{a}{b} \equiv k\...
- 56
1
vote
1
answer
49
views
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 ...
- 1,290
0
votes
0
answers
35
views
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,...
- 2,701
2
votes
2
answers
75
views
Is the unitary group of a $p$-adic anisotropic hermitian space commutative?
Let $E/\mathbb Q_p$ be a quadratic extension where $p \not = 2$. Let $V$ be an $E$-vector space equipped with a non-degenerate hermitian form, and assume that $V$ is anisotropic. In particular, $\dim(...
- 4,579
0
votes
1
answer
62
views
Strategy to prove continuity of group action $G×L→L$
Let $G$ and $L$ be topological groups, and $G$ acts on $L$ via the map $f:G×L→L$.
I want to prove $f$ is continuous. From definition,$f$ is continuous if only if for arbitrary open subset $U$ of $L$, $...
- 2,701
0
votes
2
answers
59
views
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 ...
-1
votes
0
answers
47
views
How to prove $\mathcal{O}_{\mathbb{C}_p}/ p \mathcal{O}_{\mathbb{C}_p}＝(\mathcal{O}_{\mathbb{C}_p}/ p \mathcal{O}_{\mathbb{C}_p})^p$
Let $\mathcal{O}_{\mathbb{C}_p}$ the
ring of integers of complex $p$-adic numbers
$\mathbb{C}_p$ which are defined as completion of alg closure $\overline{\mathbb{Q}_p}$ with respect $\vert \cdot \...
- 574
0
votes
1
answer
25
views
value group of $E＝\Bbb{Q}_p(p^{1/e})$
I want to find what is a value group of $E＝\Bbb{Q}_p(p^{1/e})$($e$ is positive integer, and this is totally ramified extension of degree $p$).
I know value group of $K＝ \Bbb{Q}_p$ is {$p^a$｜$a∈\Bbb{Z}$...
- 574
0
votes
1
answer
68
views
What is the value group of $\overline {\Bbb{Q}_p}$ and $ \Bbb{C}_p$ ? And are they discrete?
What is the value group of $\overline {\Bbb{Q}_p}$ and $ \Bbb{C}_p$ ?
And are they discrete?
For finite extension of $ \Bbb{Q}_p$, there are known results for extension of valuations, but what about ...
- 574
0
votes
1
answer
33
views
$(O_K/pO_K)^p＝O_K/pO_K$holds, then $∃b∈K^×$, such that $｜a-b^p｜≦｜p｜$
Let $L$ be finite extension of $ \Bbb{Q}_p$ and field $K$ satisfies $L⊆K⊆\Bbb{C}_p$.Let $O_K$ be ring of integers of $K$.
Suppose $(O_K/pO_K)^p＝O_K/pO_K$・・・① holds, then $∃b∈K^×$, such that $｜a-b^p｜≦｜...
- 574
1
vote
1
answer
56
views
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 ...
- 9,587
0
votes
0
answers
58
views
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 ...
- 574
0
votes
1
answer
58
views
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 ...
- 27
0
votes
1
answer
65
views
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 ...
1
vote
2
answers
60
views
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 ...
- 390
2
votes
0
answers
88
views
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 ?
- 390
1
vote
0
answers
41
views
Given a solution to a polynomial over $\mathbb{Q}_p$, find a solution mod $p$
Given the solution of a diophantine polynomial $f$ in $\mathbb{F}_p$, we can use Hensel's lemma to extend this to a solution in $\mathbb{Z}_p$, and the natural homomorphism $\mathbb{Z}_p \to \mathbb{Q}...
- 63
0
votes
1
answer
64
views
The order of residual field of $p$-adic field
Let $L/ \Bbb{Q}_p$ be finite extension. Let $o$ be ring of integers of $L$.Let $π$ be a uniformizer, and $q$ be order of residue field.
I want to determine the order of $(o/\pi^no)^\times $.
I know ...
- 390
0
votes
0
answers
24
views
Writing factorial $x!$ in terms of the $p$-adic gamma function?
Let $x$ be a $p$adic integer.
How can we write $x\cdot (x-1)\cdot (x-2)\cdots (x-n)$ in terms of the $p$-adic gamma function (Morita gamma function) ?
The same question for the product $x\cdot (x-2)\...
- 3
0
votes
0
answers
31
views
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 ...
- 9,587
0
votes
0
answers
45
views
Residue class field of $\mathbb{Q}_p(\zeta_m)$, where $(m,p) = 1$, is $\mathbb{F}_p(\zeta_m)$
I've been learning about local fields for some time now and I wanna prove the following:
Let $p$ be a prime number and $m \in \mathbb{N}$ such that $(m,p) = 1$. Pick a primitive $m$-th root of unity $\...
- 577
2
votes
1
answer
26
views
Understanding the Jacquet module of the Steinberg Representation
Let $G=GL_2(F)$ where $F$ is a non-Archimedean local field of characteristic $0$, for example $\mathbb{Q_p}$. Let $\chi=1_T$ be the trivial character of the maximal split torus $T=\begin{pmatrix}* &...
- 455
1
vote
1
answer
57
views
Degree of extension concerning p-adic numbers.
We define
$$\mathbb{Z}[[X]]_{conti}:=\{t:\mathbb{R}_{\geq 0}\rightarrow \mathbb{Z}\ (t\in \mathbb{Z}^{\mathbb{R}_{\geq 0}})|\forall M\geq 0\ \{r\in\mathbb{R}_{\geq0}|r\leq M\land t(r)\neq0\}\ are \ ...
- 351
1
vote
0
answers
53
views
Smooth functionals in the contragredient representation of a locally profinite group
I'm reading Bump's book Automorphic Forms and Representations and getting stuck in Section 4.2 about contragredient representation of smooth representations.
Setup: Let $G$ be a locally profinite ...
- 1,387
2
votes
0
answers
160
views
Why ring of Witt vectors does not depend on a choice of a prime element?
Let $R$ be a ring. Let $(W(R),＋,×)$ be a ring of Witt vectors. Let $L$ be a finite extension of $p$-adic field.
Let $O$ be the ring of integers of $L$, let $π$ be a uniformizer of $O$ and $q$ be the ...
- 2,701
2
votes
0
answers
29
views
Ideal determined by completions
Let $\mathcal{O}$ be an order in a number field $K$.
Question: Is specifying an invertible ideal $I$ of $\mathcal{O}$ equivalent to specifying invertible ideals $I_p$ of $\mathbb{Z}_p \otimes_{\mathbb{...
- 2,757
0
votes
1
answer
80
views
Why there is no ring hom from $\Bbb{Z}_p$ to $\Bbb{Z}$?
Why there is no ring hom from $\Bbb{Z}_p$ to $\Bbb{Z}$?
Let $f:\Bbb{Z}_p→ \Bbb{Z}$ be a ring hom.
My try:
In the case of $p＝5$, f:$\Bbb{Z}_5→\Bbb{Z}$ should take $\sqrt -1$(it exists because of Hensel ...
- 390
6
votes
1
answer
134
views
Group of units of $\mathbb{Z}_3[[x]]$
I am trying to calculate the group of units of the power series ring $\mathbb{Z}_3[[x]]$. I know that all the unit elements are of the form $u+\sum_1^{\infty} a_nx^n$ where $u$ is a unit in $\mathbb{Z}...
- 5,884
1
vote
0
answers
70
views
In an affinoid/Tate algebra, can one always find a system of topological generators that have specific properties at prescribed points?
Let $k$ be a complete non-archimedean field with absolute value $| \cdot |$.
The Tate algebra over $k$ in the variables $X_1,\ldots,X_n$ is $T_n = k\langle{X_1,\ldots,X_n} \rangle = \{ \sum_{J \geq 0}...
- 618
4
votes
1
answer
44
views
Integrating an additive character over a local field
Let $F$ be a non-Archimedean local field, $\psi$ a non-trivial additive character of $F$. Let $\mathfrak{o}$ be the ring of integers of $F$, and $\mathfrak{p}$ be the maximal ideal of $F$. Endow $F$ ...
- 1,137
0
votes
2
answers
73
views
Does $\mathbb{Q}_2(\sqrt{-1},\sqrt{10})$ contain $\mathbb{Q}_2(\sqrt{5})$?
My suspicion is yes, because I think $\mathbb{Q}_2(\sqrt{-1},\sqrt{10})$ is not totally ramified over $\mathbb{Q}_2$, so it should contain the unique unramified extension, $\mathbb{Q}_2(\sqrt{5})$. If ...
- 323
0
votes
1
answer
44
views
Calculate Hasse-Minkowski invariant
I'm trying to show (in-)equivalence of two quadratic forms and got stuck calculating the Hasse-Minkowski invariant. After diagonalisation of the quadratic form I get that $c(f)=(1,1)$ and $c(g)=(2,\...
- 19
0
votes
0
answers
30
views
Determine if the form $x_1^2 + x_2^2 -15(x_3^2+x_4^2)$ is isotropic over $\mathbb{Q}$.
I'm trying to show whether the form $x_1^2 + x_2^2 -15(x_3^2+x_4^2)$ is isotropic over $\mathbb{Q}$. I tried to apply the strong Hasse principle, but I don't get how to show isotropicity over $\mathbb{...
- 19
1
vote
1
answer
44
views
Is any finite index subgroup of multiplicative group of p-adic field open?
I found that any finite index subgroup of multiplicative group of p-adic integer is open. But i don't know how to prove that any finite index subgroup of multiplicative group of p-adic field is open ...
- 11
0
votes
1
answer
56
views
From Galois group over $\mathbb{Q}_p$ to Galois group over $\mathbb{Q}$: possible answer
I have a potential answer to my question here but I have serious doubt.
The question was the following: for a polynomial $P\in Q[X]$ we have a decomposition field $K$ and $G=\text{Gal}(K/\mathbb{Q})$. ...
- 2,516
0
votes
0
answers
26
views
To what extent do the fixed points of this isometry on natural numbers in the 2-adic space constrain the isometry?
Consider an isometric homeomorphism $T$ on the 2-adic integers $\Bbb Z_2$, which fixes the numbers $-1,0$ and exchanges $(-1/3,1)$, and obeys $2T(x)=T(2x)$.
I'm aware that, in general, p-adic ...
- 8,211
2
votes
0
answers
47
views
From Galois group over $\mathbb{Q}_p$ to Galois group over $\mathbb{Q}$
Let $P\in\mathbb{Q}[X]$ be some polynomial. Let $p$ be some prime number. We can consider $P\in\mathbb{Q}_p[X]$. What informations can we get from the Galois group of $P\in\mathbb{Q}_p[X]$ to the ...
- 2,516
1
vote
2
answers
58
views
$G$-composition length of representation
I do not understand what is meant by $G$-composition length. The textbook I'm using - The Local Langlands for GL(2) - makes the following statement.
Let $\chi=\chi_1 \otimes \chi_2$ be a character of ...
- 455
2
votes
1
answer
35
views
p-adic norm, show $|y^2 - a|_p < \epsilon$
The p-adic norm of $x$ is denoted by $|x|_p$ and defined to be $p^{-e}$ if $p^e$ is the power of $p$ appearing in prime decomposition of $x$.
Then suppose x and a are integers and $x^2 \equiv a$ mod $...
- 161
4
votes
1
answer
72
views
How to do integration of matrices on local fields?
I'm currently reading Buzzard's note and trying to calculate the integral on page 5:
$$
S(f)(\mathrm{diag}(\varpi, 1)) = q^{-1/2} \int_N f \left( \begin{pmatrix} \varpi & \varpi n \\ 0 & 1 \...
- 1,387
0
votes
0
answers
39
views
Calculating Hilbert Norm Residue Symbol
The following properties of the Hilbert Norm Residue Symbol are given:
$(a,b) = (b,a)$
$(a_1a_2,b)=(a_1,b)(a_2,b)$ (same for $(a,b_1b_2)$)
$(a,-a)=1$ for all a
We now have to reformulate i) $(a_1/b,...
- 19
1
vote
0
answers
53
views
Find 3rd root in $\mathbb{Q}_3$ using Hensels Lemma
Let $a \in \mathbb{Q}_3$ and suppose that $\vert a-1 \vert_3 \leq 3^{-2}$. Show that $a \in {\mathbb{Q}_3}^3$ using Hensel's Lemma.
My idea is the following: I consider $f(x) = x^3 - a$ and want to ...
- 19
1
vote
1
answer
65
views
Valuation of the different for totally ramified extensions
I'm trying the following two exercises from Andrew Sutherland's MIT Number Theory 1 problem sets.
Fix an odd prime $p$.
Let $L/\mathbb Q_p$ be a totally ramified extension of degree $p$ with ...
- 1,208