Questions tagged [hensels-lemma]

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

Filter by
Sorted by
Tagged with
0
votes
0answers
27 views

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(...
1
vote
1answer
24 views

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 ...
0
votes
1answer
56 views

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(...
1
vote
2answers
58 views

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 ...
1
vote
0answers
32 views

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) \...
0
votes
1answer
19 views

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 $R$ ...
1
vote
1answer
43 views

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 ...
2
votes
1answer
33 views

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,...
1
vote
1answer
50 views

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. ...
4
votes
1answer
71 views

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 $...
2
votes
1answer
117 views

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 $...
1
vote
0answers
49 views

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[...
1
vote
0answers
39 views

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 ...
1
vote
0answers
46 views

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 ...
0
votes
1answer
97 views

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 ...
2
votes
0answers
74 views

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 $\...
0
votes
0answers
24 views

Is there a product expression for non-primitive roots of Fibonacci numbers?

I found a product related to Fibonacci numbers, from T.M. Apostol 1976. $\phi_{n}=\prod_{d|n}F_{d}^{\mu(n/d)}$, for all divisors d of n. Since the last iteration above is merely the whole Fibonacci ...
0
votes
0answers
60 views

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 ...
1
vote
0answers
79 views

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} $...
2
votes
0answers
67 views

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 ...
2
votes
0answers
52 views

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^...
2
votes
1answer
108 views

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 ...
1
vote
2answers
105 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 ...
2
votes
1answer
102 views

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 ...
1
vote
1answer
91 views

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 ...
2
votes
0answers
61 views

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 ...
0
votes
0answers
52 views

Hensel's lemma for the completion of $\mathbb{F}_q(t)$

If we want to find the roots of a polynomial $f(x)$ modulo a prime $p$ to the power of $n$, we can use Hensel's lemma. Let's say we want to find all roots of $x^3+x^2+4x+1$ mod $49$. Then we can use ...
1
vote
0answers
71 views

Intuition and technique for (strict) Henselization of nodal cubic at node

Consider the union of the axes $\frac{\Bbbk[x,y]}{(xy)}$. There are two irreducible components passing through the origin which correspond to the two minimal primes of the local ring at the origin. ...
0
votes
0answers
38 views

what would be the solution by using Hensel's Lemma?, p-adic numbers

Point out the main difference or relation between Newton's polygon and Hensel's lemma when it comes to find solution of the two variable polynomial $ f(x,y)=y^6-5xy^5+x^3y^4-7x^2y^2+6x^3+x^4=0$. $ \...
1
vote
1answer
110 views

$x^2\equiv 5 \pmod{1331p^3}$

Let $p$ be given by $p=2^{89}-1$ and note that it is a Mersenne Prime. The problem is to find the number of incongruent solutions to $$ x^2\equiv 5 \pmod{1331p^3} $$ I began the problem by splitting ...
7
votes
2answers
475 views

Hensel lemma - generalization?

Let $f \in \Bbb Z[X]$ be monic and assume that $f$ has a root $a_n$ modulo $p^n$ for every $n \geq 1$ (where $p$ is a fixed prime). Does it follow that $f$ has a root in $\Bbb Z_p$? The problem is ...
1
vote
1answer
69 views

If an $\mathbb{F}_p$-point is smooth, then it has Hensel lift

In the writeup(http://www.crm.umontreal.ca/sms/2014/pdf/stoll.pdf), the following statement has been made on page 2, under the proof of Proposition 3 : If $p + 1 > 2g \sqrt p$ and $p$ does not ...
1
vote
0answers
155 views

Equivalent statements of Hensel's lemma

There exist many different equivalent formulations of Hensel's lemma in the literature. Usually the proof of their equivalence is difficult and uses deep knowledge of commutative algebra. I asked ...
4
votes
1answer
117 views

Another generalization of Hensel's lemma

I know this is a "dangerous" topic to ask a question about, since a lot of questions regarding Hensel's lemma have already been answered, but I searched for it and couldn't find this version of the ...
2
votes
2answers
183 views

Henselian field and Monic Polynomials with - Neukirch exercise 5 (Henselian Fields)

i tried to solve the following exercise: Let $K$ be a nonarchimedean valued field, $\mathcal o$ the valuation ring, and $\mathfrak p$ the maximal ideal. $K$ is henselian if and only if every ...
0
votes
1answer
265 views

Hensel's lemma for complete rings

Hensel's lemma can be stated as Let $A$ be a local ring complete by the maximal ideal $m$, and define $Q=A/m$. Let $f\in A[x]$ be a monic polynomial and define $\bar{f}\in Q[x]$ as the polynomial ...
2
votes
1answer
79 views

Hensel Lifting with a non-simple root

I was doing a problem and it involves lifting a root x= 55, from mod $2^{10}$ to a solution mod $2^{19}$ but the root is non simple, i.e. $$f'(x) \equiv 0 (mod 2)$$ Here, $f(x) = x^{3} - 9x + 8 \...
8
votes
1answer
436 views

Criteria for a cubic polynomial in $\Bbb Q[x]$ to split completely over $\Bbb Q_p$

Background: A quadratic polynomial splits over a field $k$ iff its discriminant is a square in $k$. Squares in $\Bbb R$ are just the elements $\ge 0$, and it is also quite easy to recognise squares in ...
6
votes
2answers
182 views

Hensel Lemma and cyclotomic polynomial

I'm trying to prove the following equivalence Let $p\neq 3$, then $f(X)=X^3-1$ splits completely in $\mathbb{Z}_p$ ($p$-adic integers) iff $p\equiv1 \bmod 3$. This is my attempt: first I noticed ...
1
vote
0answers
49 views

Is $\gcd(2^m-1,2^n-1) = 2^{\gcd(m,n)}-1$ related to a Hensel lift?

Is $\gcd(2^m-1,2^n-1) = 2^{\gcd(m,n)}-1$ related to a Hensel lift? They look similar in respect of having the same "modulo some power of $p$" component. Are they both examples of a "lift"?
2
votes
0answers
77 views

How can I prove corollary 7.4 in Eisenbud's commutative algebra book?

Eisenbud states the following corollary to Hensel's lemma: Given a polynomial $f(t,x)$ over a field $k$, with $x=a$ a simple root of $f(0,x)$, then there exists a unique power series $x(t) \in k[[t]]$...
4
votes
1answer
265 views

Hensel lemma for schemes and henselian rings

One version of Hensel's lemma for schemes is simply the definition of formally unramified: A scheme $X$ is said to be formally etale if: For a ring $R$ with an ideal $I$ such that $I^2 = 0$, one ...
1
vote
1answer
36 views

$\alpha \in \mathbb{Z}_p^{\times}$ iff there are infinitely many $n \in \mathbb{Z}_{>0}$ such that $X^n = \alpha$ has a solution.

Let $\alpha \in \mathbb{Q}_p^{\times}$. Then $\alpha \in \mathbb{Z}_p^{\times}$ iff there are infinitely many $n \in \mathbb{Z}_{>0}$ such that $X^n = \alpha$ has a solution. Approach: $\...
2
votes
1answer
350 views

$\mathbb{Z}_p^*$ has a primitive $p - 1$-th root of unity (p-adic)

I am trying to prove that $\mathbb{Z}_p^*$ has a primitive $p - 1$-th root of unity. I already proved that $\mathbb{Z}_p^*$ has a $p - 1$-th root of unity using Hensel's lemma. Here is my proof: if ...
0
votes
1answer
89 views

$U/U_3 \equiv (\mathbb{Z}/27\mathbb{Z})^{\times}$

I was reading this: What is the group structure of 3-adic group of the cubes of units?, but I do not understand why $U/U_3 \equiv (\mathbb{Z}/27\mathbb{Z})^{\times}$
2
votes
0answers
277 views

Squares in $\mathbb{Q}_p \bmod p$ (p-adic numbers)

I was reading the book P-adic numbers: An introduction by Fernando Gouvea and I found the following problem: Let $m \in \mathbb{Z}$, and suppose that the congruence $X^2 \equiv m\pmod p$ has a ...
4
votes
0answers
123 views

square root of $-1$ in $\mathbb{Q}_p$

In Computing the quotient $\mathbb{Q}_p[x]/(x^2 + 1)$ and Decomposition of the tensor product $\mathbb{Q}_p \otimes_{\mathbb{Q}} \mathbb{Q}[i]$ into a product of fields it has been claimed that since $...
2
votes
2answers
103 views

Computing the quotient $\mathbb{Q}_p[x]/(x^2 + 1)$ [duplicate]

I am trying to compute the quotient $\mathbb{Q}_p[x]/(x^2 + 1)$ where $\mathbb{Q}_p$ represents the $p$-adic numbers. I already proved that if $p \equiv 2,3\mod4$, then $\mathbb{Q}_p[x]/(x^2 + 1) \...
0
votes
2answers
208 views

A $p$-adic unit is an $n^{th}$ power if it is congruent to a $n^{th}$ power mod $m$ (Eisenbud exercise)

Exercise 7.27 in Eisenbud's Commutative Algebra poses the following problem: Give a criterion for a $p$-adic unit $u$ to be a $n^{th}$ power for any $n$. The hint states: "The form of the criterion ...
0
votes
1answer
70 views

Computing $m^n (mod\,p^2)$ efficiently for a large prime $p$

As the title says, I want to know if there's a fast way to compute $m^n (mod\,p^2)$ for some large prime $p$. Obviously, I can compute $pp = p^2$ and then just use an exponentiation algorithm to ...