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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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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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 ...
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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 ...
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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\...
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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}...
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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 ...
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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 $0$...
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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\...
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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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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(...
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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 $L$, $...
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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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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 \...
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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 {$p^a$|$a∈\Bbb{Z}$...
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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$, there are known results for extension of valuations, but what about ...
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$(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|≦|...
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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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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}...
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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 ...
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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)\...
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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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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 $\...
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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}* &...
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1 answer
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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 \ ...
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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 ...
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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 ...
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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{...
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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 ...
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6 votes
1 answer
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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}...
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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}...
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4 votes
1 answer
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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 $F$ ...
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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 ...
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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,\...
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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{...
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1 vote
1 answer
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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 ...
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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 $G=\text{Gal}(K/\mathbb{Q})$. ...
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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 ...
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2 votes
0 answers
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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 ...
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1 vote
2 answers
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$G$-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 ...
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  • 455
2 votes
1 answer
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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 $...
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1 answer
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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 \...
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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,...
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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 ...
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1 vote
1 answer
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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 ...
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