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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70 views

Adjoing $p$th roots to rings: as quotients of polynomials

Motivation/Setup: Suppose $R$ is a domain. I'd want to "adjoin $p$th roots of an element $a \in R$." where $p$ is a prime (which i am mostly interested in $p$-adic) This seems to be a cmomon ...
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Is $|\mathbb Z_p/(p-1)\mathbb Z_p|=1$?

Let $\mathbb Z_p$ denote the ring of p-adic integers. We know $\mathbb Z_p/p\mathbb Z_p\simeq\mathbb F_p$. And then I met a problem asking what is $\mathbb Z_p/(p-1)\mathbb Z_p$. I think there is only ...
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Constructing convergent sequences in p-adic numbers

I am trying to find a prime p such that there exists a sequence of integers S_n where (S_n)^2 converges to 21 under the p-adic norm. I'm familiar with the definition of p-adic distance and with the ...
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$1+px^2$ is a square iff $x\in\mathbb Z_p$

I'm fairly new to number theory and have just started to study p-adic numbers. In a book I'm reading about the fundamentals of number theory, there is the following statement that is supposed to be ...
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Why is $2$-adic roots of unity $\{1,-1\}$

I am reading a proof of the fact that the only root of unity in $\mathbb Q_2$ is $\pm1$. But I am stuck at one point: The proof says that all of the root of unity in $\mathbb Q_2$ has an order of a ...
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Why is $\operatorname{Aut}(\mathbb{Z}_p)\cong \mathbb{Z}_p^*$?

Let $\mathbb{Z}_p$ denote the $p$-adic integers. Why is $\operatorname{Aut}(\mathbb{Z}_p)\cong \mathbb{Z}_p^*$? I know that $\operatorname{Aut}(\mathbb{Z}/p^n\mathbb{Z})\cong(\mathbb{Z}/p^n\mathbb{Z})^...
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Converse to Hensel's Lemma

The following was an exercise in some MIT course notes on p-adic numbers and Hensel's lemma. Let $f\in \Bbb Z_p[X]$. Suppose that $b$ is a simple root of $f$. Prove that for any $a\in \Bbb Z_p$: if $|...
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Is $\mathcal{O} \otimes_{\mathbb{Z}_p} F$ a finite direct product of fields?

Consider the $p$-adic field $\mathbb{Q}_p$ with ring of integers $\mathbb{Z}_p$ and residue field $\mathbb{F}_p$. Let us take a finite extension $K$ of $\mathbb{Q}_p$, with ring of integers $\mathcal{...
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Simple proof to show $\mathbb{Q}_2((-2)^{\frac{1}{2}})$ is contained in a cyclotomic extension

Let $(-2)^{\frac{1}{2}}$ be any element in $\overline{\mathbb Q_2}$ satisfying $x^2+2=0$, then is there a simple way to show that $\mathbb{Q}_2((-2)^{\frac{1}{2}})$ is contained in a cyclotomic ...
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Is the limit of a squence of p-adic integer also a p-adic integer?

Given a squence of p-adic integers $\{x_n\}$, and assume $\lim_{n\to\infty}x_n\in\mathbb Q_p$ exists in the sense of p-adic metric. Do we know the limit is also a p-adic integer, i.e. $\lim_{n\to\...
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Is it always possible to construct an extension of $\mathbb{Q}_p$ as a totally ramified extension first and then an unramified extension after?

Let $K = \mathbb{Q}_p$, and $L/K$ be a finite Galois extension. Question: Is it possible to find a totally ramified extension $L'/K$ such that $L/L'$ is unramified? I know that it is always possible ...
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Polynomial close to separable polynomial in Gauss norm is separable itself.

Let $(K,|\cdot |)$ be a complete non-archimedean valuation field. Let $f(x)\in K[x]$ be a separable polynomial of degree $n$. Let $||\cdot||$ be the Gauss norm associated with $|\cdot|$, i.e. $||f||=\...
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What are the bounds on the frequency of divisions by 2 in the Collatz cycles of natural numbers?

If $T^n(x):\Bbb N\times\Bbb N\to\Bbb N$ is the $n^{th}$ iterate of the Collatz function $T(x)=\frac12(3x+1)$ if $x$ odd and $x/2$ if $x$ even. Then let $Q(x):\Bbb N\to2^{\Bbb N}$ be the parity ...
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Small linear combination of $p$-adic numbers

Let $c_1,...,c_s$ be nonzero $p$-adic numbers. Does a positive integer $C$, depending only on $c_1,...,c_s$, and with the following property: For all $p$-adic numbers $a_1,...,a_s$ and integer $n\geq ...
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Monotonic functions bijectively mapping rational numbers to $n$-adic rationals (numbers having finite expansions in base $n$)

The Minkowski's question-mark function is a monotonic bijective function from rational numbers to numbers with finite binary representations. For which natural numbers $n$ is there a monotonic one-to-...
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Is it possible to construct a $p$-adic numbers out of $\mathbb{Z}[[x]]$ like we can for the real numbers?

I saw this question earlier today about the $p$-adic numbers generally and am trying to figure out how to visualize them or construct an algebraic entity with the right structure. There's a ...
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p-adic numbers vs real numbers

Could anyone give a concrete example of a p-adic number that is not a "real number"? that is, do we create "new numbers" (non real numbers) by completing Q with a non Archimedean ...
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Proving the sums of three cubes conjecture by the Hasse principle

In his Cours d'arithmétique Serre applies the Hasse-Minkowski theorem to quadratic forms of the form: $$ x^2 + y^2 + z^2 = n $$ for $n \in \mathbb{N}$ to prove that a natural number $n$ is a square if ...
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Are $p$-adic groups also Lie groups? What is a $p$-adic group specifically? [duplicate]

I am a bit confused by the definition of a $p$-adic group. I initially thought that a $p$-adic group was simply a group with $\mathbb{Q}_p$ as the underlying manifold, however this seems to be too ...
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Evaluate $ \int_{\mathbb{Q}_p} |x^2|_p \, dx $ with respect to Haar measure on $\mathbb{Q}_p$

Is it possible to do calculus problems over the $p$-adic numbers, $\mathbb{Q}_p$ ? Let $d\mu = \frac{dx}{|x|_p}$ be the Haar measure on $\mathbb{Q}_p^\times$. What would be the value of $$ \int_{\...
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Has a dyadic logarithm been studied?

Define a dyadic logarithm of an odd dyadic integer $a$ as the limit of $\frac{a^z-1}{z}$ for $z\to0$ in the dyadic sense. I can prove, I'm assuming it's also well known, that dyadic exponentiation ...
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Corollary of Kronecker-Weber Theorem (J. Neukirch's ANT)

I have a question about the proof of Corollary (1.9), Chap V page 324 from Jürgen Neukirch's Algebraic Number Theory: Claim: Every finite abelian extension of $L \vert \mathbb{Q}_p$ is contained in a ...
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What about the algebraic structure of $T_p(G) \otimes_{\mathbb{Z}_p}A$?

Let $G$ be a $p$-divisible group over the ring of $p$-adic integers $O_K$ of $p$-adic field $K$. The $p$-adic Tate module $T_p(G)$ of $G$ is rank $1$ free $\mathbb{Z}_p$-module. Then $T_p(G) \otimes_{...
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Obtaining the composite field of two p-adic fields in MAGMA

Let $K= \mathbb{Q}_3$ and $L = \mathbb{Q}_3(\alpha)$ be defined by $\min_K(\alpha) = x^4 - 3x^2 +18$. Furthermore, let $F$ be the unramified extension of $K$ of degree $4$ (which is generated by a ...
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Implementation in MAGMA: Field extension over the p-adics with a polynomial which is neither inertial nor Eisenstein

Let $K = Q_3$ and $L = K(a)$ be the extension of $K$ defined by the polynomial $f = x^6+3x^5-2$ (i.e. this is the minimal polynomial of $a$ over $K$). Now I would like to obtain this field $L$ in ...
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A p-adic Fourier transform

Consider the field of $p$-adic numbers $\mathbb Q_p$. Define the character $\chi(u p^n) = e(p^n)$ for all $n \in \mathbb Z$ and all unit $u$. In particular it is trivial on integers. This allows to ...
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$L^1$ norm of a function is greater than the norm of its root

Edit: My friend helped me to solve this, and now it is solved for me. We must first separate the simple cases, the only serious case is when $\sum_{i=0}^n |f_i|$ and $\alpha$ are both strictly greater ...
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Explicitly determine the prime ideal such that $\mathbb{Q}_5(\sqrt{5})$ is the completion of some number field

Let $K = \mathbb{Q}_5(\sqrt{5})$. As I learned from my last post, it is possible to write $K$ as the completion of $(k,v)$ where $k$ is a number field (i.e. a finite extension of $\mathbb{Q}$) and a ...
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Can a generator of the ring of integers of local fields can be chosen so that it is also a uniformizer at the same time?

Let $L/K$ be an extension of local fields. We can find $\alpha$ such that $\mathcal{O}_L=\mathcal{O}_K[\alpha]$. What do we know about this generating element? I think that this $\alpha$ can be ...
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Finite extensions of $\mathbb{Q}_p$ as completions of number fields with some valuation over a prime ideal [duplicate]

In my first post, I asked if $\mathbb{Q}_p$ is the completion of $\mathbb{Q}$ over some valuation of an prime ideal (and it seems to be true, according to an answer). Now I am asking if this can be ...
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Understanding the derivation of the $p$-adic numbers as the completion from $\mathbb{Q}$ wrt. the prime ideal $(p)$

Since I just started to learn algebraic number theory and feel quite insecure about the subject, I apologize in advance for any wrong notations or conclusions. I tried to write down what I understood ...
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what is $\mathbb{Q}_p^\times/\mathbb{Q}_p^{\times^2}$ called?

I have to do a presentation about $p$-adic numbers and I don't know what to call $\mathbb{Q}_p^\times/\mathbb{Q}_p^{\times^2}$. Can you help me, please? Thank you!
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Galois theoretic way to find which completions of a number field are isomorphic

Question: This is not really a question because I think I have a solution, so I am asking for a review, some opinion for improving and going further, for solving the ambiguous points, as well as a ...
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Definition of $\Lambda$-adic cusp form

Definition 4.2.2 of https://www.math.arizona.edu/~swc/aws/2018/2018SharifiNotes.pdf defines a $\Lambda$-adic form (where $\Lambda$ is an Iwasawa algebra) to be a cusp form if all but finitely many of ...
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Determining the corresponding local field

Let $p$ be an odd prime number. Then there are three quadratic extensions of $\mathbb{Q}_p$. Assume that $t$ is a nonresidue module $p$, then these three extensions can be obtained by adjoining $\sqrt{...
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Convergence of sequence in p-adic integers

Let $a\in Z_p^{*}$. I need to prove the sequence $\{a^{p^n}\}_{n\geq 0}$ converges in $\mathbb{Z}_p$ to $w$, where $$a=wb$$ for $w$ is the $p-1$ root of unity in $\mathbb{Z}_p^*$ and $b\in 1+p\mathbb{...
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Convergence of series in p-adic norm

I want to know the convergence of the series $\sum_{n\geq 0} {{p^{n+1}}\choose{p^n}}$ My idea is to show the convergence of the partial sum $|s_n-s_{n-1}|_p$ but I am stucked in expressing the terms ...
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Ultrafilters on naturals determine p-adic numbers

While preparing for a short lecture on ultrafilters for undergraduates, I realized some interesting things I have never read about. Though I'm asking now a specific question, any reference about this ...
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Determine a generator of a subextension in a non-Kummer-setting

Let $K = \mathbb{Q}_3$ and $L=K(\alpha)$ where $\min_K(\alpha) = x^4 - 3x^2 + 18$. Furthermore, let $F/K$ be the unique unramified extension of degree $4$. It can be shown that $F/K$ is generated by a ...
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Prove: If $a \in \mathbb{Q}^*$ has a square root in $\mathbb{Q}_p$ for all primes $p$, then it has a square root in $\mathbb{Q}$

I'm seeking to prove or disprove the following statement: If $a \in \mathbb{Q}^*$ has a square root in $\mathbb{Q}_p$ for all primes $p$, then it has a square root in $\mathbb{Q}$. (It is not assumed ...
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Proof that all extensions of $\mathbb{Q}_p$ are of the form $\mathbb{Q}_p[\sqrt[n]{a}]$

I'm going over some notes that claim that any extension of $\mathbb{Q}_p$ of degree $n$ has the form $\mathbb{Q}_p[\sqrt[n]{a}]$ for some $a\in\mathbb{Z}_p$. It references Proposition III.12 from ...
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How to show that any nontrivial valuation on the field of rational numbers is equivalent to some p-adic valuation?

I have started with a nontrivial valuation v on Q. Suppose V is the corresponding valuation ring. I want to show that V is equal to the valuation ring of some p-adic valuation for some prime p. Then ...
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Over which extensions of $\mathbb Q_2$ is $X^2+Y^2+Z^2$ isotropic?

I was playing around a bit with quadratic forms for a different question here, and among other things had to decide when the form $X^2+Y^2+Z^2$ is isotropic over a certain field $k$, i.e. whether ...
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On measures on profinite groups

A measure on a profinite group $\Gamma$ with values in a $p$-adic ring $\cal O$ and its reinterpretation as an element of the Iwasawa algebra $\Lambda_{\cal O}={\cal O}[[\Gamma]]$ are defined in ...
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5-adic numbers represented by binary quadratic form

This question comes from Borevich-Shafarevich Number Theory CH.1 Sec.6, from which I am teaching myself about p-adic numbers. It asks you to find all 5-adic numbers represented by the form $f = 2x^2 + ...
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Inertia degree of primes in p-adic extensions

I'm reading through some number theory and ran across a theorem where the proofs referenced were incomprehensible to me, and I was hoping there might be a simpler proof than slogging through another $\...
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Hasse-Minkowski for cubic forms

We know that an analogue of the Hasse-Minkowski theorem does not hold for all cubic forms, e.g. because Selmer's cubic: $$ 3x^3 + 4y^3 + 5z^3 = 0 $$ has solutions over $\mathbb{R}$ and $\mathbb{Q}_p$ ...
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55 views

Identification of ends of Bruhat-Tits Tree

I am trying to understand why a canonical identification exists between the "ends" or "rays" of the Bruhat-Tits tree defined on $\mathbb{Z}_p$-lattices (with metric invariant under ...
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What is the minimal polynomial of an $8$-th primitive root of unity over $\mathbb{Q}_3$?

Let $K = \mathbb{Q}_3$ and $\zeta_8$ a primitive $8$-th root of unity. Question: What is $\min_K(\zeta_8)$? I know that the extension $K(\zeta_8)/K$ is unramified of degree $2$, so the degree of the ...
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Introduction to p-adic numbers

I am a freshman and for a final project of a subject I have to give an introduction to p-adic numbers, I look for some sources (books, videos, articles) to be able to do my work, the only bases I ...

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