Questions tagged [hensels-lemma]
For questions regarding Hensel's lifting lemma in modular arithmetic and its generalization to commutative rings.
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Ring isomorphism via Hensel lifting
Consider a monic polynomial $f\in\mathbb Z[X]$ (see Note). Assume that we can factor $f\bmod p=gh$ into two monic irreducible polynomials $g\neq h\in\mathbb F_p[X]$ of same degree (or more generally, ...
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Applying Hensel's lemma to solve $x^2 + 8 \equiv 0\pmod {121}$. [duplicate]
When solving for $x^2 + 8 \equiv 0 \pmod {121}$, How can we apply Hensel's lemma to solve for its solutions? What I currently understand is that for a prime $p$ and $e \geq 2$, then $f(x) \equiv 0 \...
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$\underset{\bar{a},\bar{b}\in S^n}{Pr}[f(x,\bar{a}t+\bar{b})\text{ is reducible and }f(x,\bar{b})\text{ is square-free}]\leq\frac{7d^6}{|S|}$
Let $\mathbb{F}$ be a field. Let $S\subset \mathbb{F}$ be a finite set
with a size large enough. Let
$f(x,y_1,y_2,\dots,y_n)=f(x,\overline{y})$ be a almost monic and
irreducible polynomial in total ...
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Lifting solutions mod $p \in \mathbb{P}$ to solutions mod $q = p_k = ({p_{k-1}})^2 + p_{k-1} + 1$ prime for some $k$
Suppose we have solutions $(x,y)$ for $x^2 - y^2 \equiv N \pmod p$ where $p \in \mathbb{P}$ or $p$ is a prime power and we want to lift these solutions modulo $q$, a prime where $$p_k = ({p_{k-1}})^2 +...
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Henselisation of Normal local rings (in Milne's Etale Cohomology)
The usual way to define the Henselisation $A^h$ of a local ring $(A, \mathfrak{m})$ is to take direct limit $\varinjlim (B, q)$ over all etale neighborhoods of $A$
(i.e. pairs $(B,q)$ where $B$ is an ...
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Composition equations of power or formal series
I am given a power series $T(z) = t_nz^n +$ some higher order term (of degrees not being multiples of $n$), $n>1$ and all $t_i \in \mathbb{N}$. I am trying to solve for $z$ in the unit disc (...
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Show irreducibility for Polynomials of degree 4 over p-adic Fields
As I'm new into p-adic Field Theory, I learned Hensel's Lemma by which I can show if a polynomial has roots over the p-adic numbers (i.e. in $\mathbb{Q}_2$ or $\mathbb{Q}_3$) . For polynomials of ...
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Number of solutions $x \in \{1,2,\ldots, 1000 \}$ of $ x^2(x+1)^2 \equiv 0 \pmod{1000}$
Since $1000 = 8\cdot 125$, I have calculated that $\lambda_f(8)=4$.Using the Hensel's Lemma, I found that $\lambda_f(125)=5$. However, it requires somewhat extensive calculations.
Is there a faster ...
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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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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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Hensel's Lemma for $f(x)= x^5 + 2x^4 + 3x^2 -x + 887$
Im trying to show that $f(x)= x^5 + 2x^4 + 3x^2 -x + 887 =0 $ mod $31^k, k\geq 2$ such that $x\equiv 2$ mod 31 has many solutions.
For now I was trying to play around with $k=2$. I know $f(2) \equiv ...
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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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On Hensel lifting of a particular quadratic polynomial mod $q^2$ where $q$ is a prime
I have a quadratic polynomial $f(x)\in\mathbb Z[x]$. I want to compute its roots modulo $q^2$.
The polynomial $f(x)\bmod q$ has a double root at $r_1$ because $disc\equiv0\bmod q$.
In my case $q|f'(...
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Lift roots of a polynomial from $\mathbb{F}_p$ to $\mathbb{Q}_p$
I was trying to prove a counterexample to Hasse's principle. To do this, I was trying to prove that the equation $x^4-17=2y^2$ has no solutions on $\mathbb{Q}$ but has solutions in $\mathbb{Q}_p$ for ...
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Application of Hensel's lemma to $\mathbb{Z}_3[[j]]$
I am trying to use Hensel's lemma to find all the elements which have a square root in $R=\mathbb{Z}_3[[j]]$. We consider $f(x)=x^2-\sum_{i=0}^{\infty}a_ij^i$ a monic polynomial in $R[x]$. Looking at ...
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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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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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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 $r$ ...
user841891
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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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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 $a$...
user799688
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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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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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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 ...
user158047
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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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