Questions tagged [hensels-lemma]

For questions regarding Hensel's lifting lemma in modular arithmetic and its generalization to commutative rings.

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Equivalent characterizations of Henselian Rings (Theorem 4.2 in James Milne's "Étale Cohomology")

I am stuck on a step in the proof of Theorem 4.2 in Chapter I of James Milne's "Étale Cohomology". The particular implication is (c) $\Rightarrow$ (d). Let $X=\text{Spec} (A)$, where $A$ is ...
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Silverman Hensel's Lemma for Selmer and Sha

I am reading Silverman's Arithmetic of Elliptic Curves. In Section X.4 (The Selmer and Shafarevich-Tate Groups) Silverman derives a diagram relating the cohomology of the elliptic curve E over the ...
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Ambiguity in definition of a henselian ring?

Let $R$ be a local ring, and $S = \operatorname{Spec} R$ be its corresponding scheme. Let $s \in S$ be the closed point. Then Bosch, Lütkebohmert and Raynaud define in Néron Models Definition The ...
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Solving congruences with cubics

I want to be able to solve $$x^3 +6x^2+x+5 \equiv 0\mod{13^2} $$ I have used Hensel's Lemma, and currently have: $ f(1+13t_1) \equiv 0\mod{13^2} $ is equivalent to $ 16t_1+117t_1^2+169t_1^3 \equiv -1\...
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Using Hensel lemma to find roots in valuation rings

Let $K$ be a p-adic field with absolute Galois group $\varGamma$. Let $O^{ur}_K$ be the ring of integers of $K^{ur}$. Then, for $n$ prime to $p$, the $\varGamma$-module $\mu_{n}$ is unramified. Hensel ...
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Two versions of Hensel's lemma

In textbooks on algebraic number theory and $p$-adic numbers, I quite often find two different statements and they are all called Hensel's lemma or at least different versions of Hensel's lemma. The ...
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Question about Hensel's Lemma

A basic version of Hensel's Lemma states: Suppose that $f(x) \in \mathbb{Z}[x]$, and integer $k \geq 2$, and $p$ is a prime, and $r$ is a solution of the congruence $f(x) \equiv 0 \mod p^{k-1}$. If $...
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other multivariable version of Hensel's Lemma

In advance, sorry for my english. In this notes there's a multivariable version of the $|f(a)|_p<|f'(a)|_p$ version of Hensel's Lemma. I tried to adapt it to a proof for a multivariable version of ...
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Multivariate Hensel's Lemma

Suppose we have a single variable polynomial $f\in \mathbb{Z}_p\subset \mathbb{Q}_p$ and we have its root $\alpha$ over the finite fields $\mathbb{F}_p$, $f(\alpha)=0$ mod $p$. By Hensel's Lemma we ...
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Hensel's Lemma for zero ring

Hansel's Lemma for an arbitrary ring is on pg. 117 of The Arithmetic of Elliptic Curves by Joseph H. Silverman: Here, when $I=R$, $R$ is automatically zero ring, could you tell me why then Hensel's ...
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50 views

Why we assume henselian ring is local?

Henselian ring is defined as local ring in which hensel lemma holds. Why do we assume local ring? What is wrong with the definition that 'henselian ring is defined as hensel lemma holds' ?
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Henselian ring which is not complete

Henselian ring is defined as a local ring which has the property that Hensel's lemma holds. I understand that completeness is sufficient condition to be Henselian, but is not necessary condition. So, ...
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How do you solve quadratic congruences with unknown modulo using Chinese Remainder Method and Hensel's Lemma

Show that for all positive integers $n$, the following congruence has solutions: $$(x^2-2)(x^2+7)(x^2+14) \equiv 0 \pmod{n}$$ I need to use the Chinese Remainder Theorem and Hensel’s Lemma. So far I ...
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Equivalence of generalized Hensel's lemmas

Let $(A,\mathfrak{p})$ be a local ring complete in the The usual version of Hensel's lemma states that if $f(a)\equiv 0\bmod{\mathfrak{p}}$ and $f'(a)\not\equiv 0\bmod{\mathfrak{p}}$, then there is ...
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79 views

If $p$ is prime, then $x^p − 2 \equiv 0 \pmod{p^k}$ has a solution for $k \ge 2$

If $p$ is prime, then $x^p − 2 \equiv 0 \pmod{p^k}$ has a solution for $k \ge 2$. I'm supposed to either prove or disprove the statement above using Hensel's Lemma. So far what I have is assuming $r$ ...
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66 views

Is $\frac{10}{459}$ a cube in $\Bbb{Q}_3$?

Is $\frac{10}{459}$ a cube in $\Bbb{Q}_3$, the 3-adic numbers? My general strategy for showing if something is a cube in $\Bbb{Q}_3$ has been: Use Hensel's Lemma to show it is (1) If it isn't, ...
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1answer
159 views

Hensel's lemma exercise

I read this thread and the author starts by stating a problem: Suppose $p>2$ is a prime. Show that $x^p+y^p=z^p$ has a solution in $\mathbb{Z}_p^{\times}$ if and only if there exists an integer $a$...
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Proof of 'stronger' Hensel's Lemma

I'm trying to understand the proof of the following statement: Let $f\in\mathbb{Z}_p[X]$ be a polynomial with coefficients in $\mathbb{Z}_p$. Suppose there is a $p$-adic integer $\alpha_1$ such that $...
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Prove that 4 is not a 3-adic cube

I'm trying to show that 4 is not a 3-adic cube. On the Wikipedia page for Hensel's lemma (https://en.wikipedia.org/wiki/Hensel%27s_lemma) I read that: "4 is not a 3-adic cube since it is not a ...
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Using Hensel's Lemma to find the number of elements satisfying a congruence

My problem is really a conceptual one, rather than a specific one, but I'll provide an example question to illustrate where my difficulty lies. This is in an exercise set provided by my professor. ...
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91 views

A (simple) polynomial congruence to modulus prime power

Take $R,n\in \mathbb Z$ and $p$ a prime. The congruence \[ x^n \equiv R\text { mod }(p)\] has $\ll _n1$ solutions $x\in \{ 0,1,...,p-1\} $ by Lagrange's Theorem. Is the same true if I replace $p$ by ...
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116 views

Local strictly henselian $\mathbb{Q}$-algebras (i.e. "points in étale topology")

In the étale topology, we have an equivalence of categories between the category of fiber functors on the (small) étale site $Ét(\text{Spec}(S))$ and the category of local strictly henselian $S$-...
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Over a henselian field, are any two norms on a given finite dimensional vector space equivalent?

Let $k$ be a field with a non-archimedean absolute value. If $k$ is complete, then two norms on a finite dimensional $k$-vector space are always equivalent. This fact is for example commonly invoked ...
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Hensel's lemma requires the valuation to be discrete

In Neukirch's Algebraic Number Theory, the formulation of Hensel's Lemma (Proposition 4.6 in Chapter II) does not require the valuation to be discrete, only nonarchimedean (unless I somehow missed the ...
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Given $a^2 \equiv n\pmod q$ find $b$ such that $b^2 \equiv n\pmod {q^2}$

Given $a^2 \equiv n\pmod q$ find $b$ such that $b^2 \equiv n\pmod {q^2}$ $a,n,q$ are given. How to find $b$? I know I am supposed to use Hensel's lemma and "lifting" $q$, I just don't know ...
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Questions about Hensel's lemma.

I was reading through Hensel's lemma, highlighted here on page $8$ and had a couple of queries: In the proof when the author wants to show that $f(\alpha)=0$; it says something like "since $f(...
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Show that the sequence constructed in the proof for Hensel's lemma is Cauchy.

I am reading up a proof of Hensel's lemma over the $p$-adic integers and found a proof here, starting on page $8$ which I feel is the most accessible...well sort of anyway. My groundings on analysis ...
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Find the number of incongruent solutions

Let $p$ be a prime number. Find the number of incongruent solutions of $$ x^{p^5}-x+p\equiv0\mod p^{2020}.$$ Let $f(x) = x^{p^5}-x+p$. Because of $f '(x)$ different from zero mod $p$. Then I say $$f(...
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Application of Hensel's lemma $x^2 \equiv a (\mod 2^L)$

Let $a$ be an odd integer. And $L \geq 1$. I would like to know the number of solutions modulo $2^L$ to the congruence $$ x^2 \equiv a \pmod {2^L}. $$ Is it possible to conclude that there is number ...
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What is the $p$-adic density of irreducible polynomials congruent to $g(x)^n$ modulo $p$?

Let $n \geq 1$ be an integer, and let $g(x) \in \mathbb{F}_p[x]$ be monic and irreducible. What is the $p$-adic density in $\mathbb{Z}_p[x]$ of monic irreducible polynomials $f(x)$ such that $f(x) \...
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Finite ring extension

in reading the proof of https://stacks.math.columbia.edu/tag/04GG (10 implies 1) I came across the following. For $R$ a local ring with residue field $\kappa$. Let $f$ be a monic polynomial over $R$ ...
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Square roots in $\mathbb Z_p$

Using Hensel's lemma, it's standard fair to show that if $p$ is odd, $d\in\mathbb Z$, and $(p,d) = 1$, then $\sqrt{d}\in\mathbb Q_p$ if and only $d$ is a quadratic residue mod $p$. Naturally, there ...
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Hensel's Lemma in $\mathbb{C} (\!( s )\!)$

Reading a proof about regular $S_n$-extensions I got stuck here: Let $g(s,X)=X^n+(s-\frac{n^n}{(1-n)^{n-1}})X+(s-\frac{n^n}{(1-n)^{n-1}}) \in \mathbb{C} (\!( s )\!)$ Specializing at $0$ we get $$g(0,...
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Let $f(x)=x^3+x^2-5$. Show that for $n=1, 2,3, ...$ there is a unique $x_n$ modulo $7^n$ such that $f(x_n)\equiv 0\pmod{7^n}$.

My gut feeling for solving this problem is to use strong induction. Starting with the base case $n=1$ we can check each of the seven congruence classes and find that $x_1=2$ is the unique solution. ...
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If $y$ is a $k$th power modulo $p^\gamma$, then it is also a $k$th power modulo $p^t$ for $t \geqslant \gamma$

This question is the true version I wanted to ask of this question. Say $p$ is an odd prime number, $k$ a positive integer and $p^{\tau} || k$. Let $\gamma = \tau + 1$. I would like to prove If $...
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Square root of $-1$ in the ring $\mathbb{R}[x]/\langle(x^2+1)^2\rangle$

Does there exist a square root of $-1$ in the ring $\mathbb{R}[x]/\langle(x^2+1)^2\rangle$? Now, any element in the ring $\mathbb{R}[x]/\langle(x^2+1)^2\rangle$ is of the form $(a+ib)+e(c+id)$ where $...
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Henselization, localizations and diagonals: are these two rings the same?

Let $k$ be a field (of characteristic zero if necessary, algebraically closed if necessary) and let $z\in k$. Let $\phi:k[T]\rightarrow k[X,Y,...]$ be the $k$-morphism given by $T\mapsto X+Y$. Let $k[...
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Hensel's Lemma - Unique in what sense?

From Wikipedia entry on Hensel's Lemma (paraphrased): if ${\displaystyle f(r)\equiv 0{\bmod {p^{k}}}\quad {\text{and}}\quad f'(r)\not \equiv 0{\bmod {p}}}$, then there exists an integer s ...
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Statement of a general form of Hensel's lemma

Let $R$ be a complete ring with respect to ideal $I$ and $p \in R[x]$ polynomial. Element $a \in R$ is an approximate root of $p$ if $$ p(a) \in p'(a)^2 I. $$ One of the general forms of Hensel's ...
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On the prime spectrum of completion of local rings

Let $(R, \mathfrak m)$ be the henselization of the local ring $\mathbb C[x,y]_{(x,y)}$ . Let $\hat R$ be the $\mathfrak m$-adic completion of $R$. Then there is a natural map $R \to \hat R$ which ...
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Strict Henselization of $k[T]_{\mathfrak{p}}$?

Let $k$ be a field and let $T$ be an indeterminate. The "points" of $\mathbb{A}^1_k=\mathtt{Spec}(k[T])$ for the etale topology are given by the strict Henselizations of the usual local rings $\...
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Is there an example for the Greenleaf theorem?

I'm looking for an example for the Greenleaf theorem: Let $f_1(x)=\dots=f_n(x)=0$ be polynomials in $\mathbb{Z}[x]$. For all except finitely many primes $p$, all solutions in $\mathbb{F}^n _p$ can ...
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Building the Henselization from above: J.S. Milne, Étale Cohomology (PMS-33), Exercise 4.9

In the book Étale Cohomology James Milne defines Henselization for a local Noetherian ring by the universal property: Let $i: A \to A^\text{h}$ be a local homomorphism of local rings; $A^\text{h} $...
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Applying Hensel's Lemma When Leading Coefficient is Not a Unit

I'm familiar with Hensel's Lemma in the case where the polynomial under consideration is monic (or has invertible leading coefficient), but I'm trying to understand how it works in the case where the ...
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Proving the number of solutions to an equation in $p$-adic numbers

I'd like to show that the equation $x^3+5x+1=0$ has exactly one solution over $\mathbb{Q}_7$, i.e., the 7-adic numbers. By Hensel's lemma, one sees that the quation has at least one solution since $1^...
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Number of solutions of a polynomial in $p$-adic integers

I want to determine the number of solutions of $f(x)=x^{19}-3x+2=0$ over $\mathbb{Z}_{19}$ and $\mathbb{Z}_{17}$ ($p$-adic integers). Is the following strategy correct for $\mathbb{Z}_{19}$ (and ...
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315 views

Count number of roots of polynomial modulo prime power

I found this problem in a number theory course, I am assuming (but not sure) it is supposed to be an application of Hensel's lemma. For every $n \in \mathbb{N}_0$, determine the number of solutions ...
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Lifting solutions modulo $2^{10}$ to a solution $2^{19}$

We are given that $55$ is a solution to $x^ 3 − 9 x + 8 \equiv 0 \pmod {2^{10}}$. Find a solution to $x^ 3 − 9 x + 8 \equiv 0 \pmod {2^{19}}$ that is a lift of $55$. I was going to try lift the ...
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Looking for this article by Paulo Ribenboim

While going through a forum post (http://mathforum.org/kb/message.jspa?messageID=40112), I found the following paper mentioned: 87a:12014 12J10 13A18 Ribenboim, Paulo (3-QEN) Equivalent forms of ...
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Every finite integral extension of a Henselian, pseudo-geometric and analytically normal ring is algebraically closed in its completion.

In the book Local Rings Nagata states in Theorem 44.1: If $R$ is a Henselian pseudo-geometric analytically normal ring, then every finite integral extension $R'$ of $R$ is analytically ...