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Questions tagged [hensels-lemma]

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

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

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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 \...
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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 ...
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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 ...
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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"?
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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 $x(t) \in k[[t]]$...
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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 ...
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$\alpha \in \mathbb{Z}_p^{\times}$ 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: $\...
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$\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 ...
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$U/U_3 \equiv (\mathbb{Z}/27\mathbb{Z})^{\times}$

I was reading this: What is the group structure of 3-adic group of the cubes of units?, but I do not understand why $U/U_3 \equiv (\mathbb{Z}/27\mathbb{Z})^{\times}$
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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 ...
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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 $...
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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) \...
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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 ...
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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 ...
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Deducing Hensel's lemma for polynomials

Can one somehow deduce the following theorem: Hensel's lemma, second version: let $f(x) \in \mathbb Z_p[x]$ be a polynomial and assume that there exist polynomials $g_1(x), h_1(x) \in \mathbb Z_p[x]$ ...
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Finding a rational solution to a polynomial with coefficients in $\Bbb{R}(t)$.

I'm at a loss trying to determine whether the very specific polynomial $$P(X):=-4X^3+tX^2+18tX-t(4t+27)\in\Bbb{C}(t)[X],$$ has a root in $\Bbb{C}(t)$. I tried using Hensel's lemma starting with the ...
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When can an algebraic number be approximated by a $p$-adic number?

Let $F$ be an algebraic function field in one variable over the finite field $\mathbb{F}_{p}$. In particular, $F$ is not perfect. Let $a \in F-F^p$ and $$f(Y)=Y^p - a \in F[Y]$$ be a purely ...
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How to use Hensel lemma to show that primitive root mod $p$ gives primitive root mod $p^2$ of the form $g + tp$

How to use Hensel lemma to show that primitive root mod $p$, where $p$ is prime, gives primitive root mod $p^2$ of the form $g + tp?$ I tried to start with congruence $g^{p-1} \equiv 1 \pmod p,$ so $...
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Tonelli and Hensel Lemma

Find 2 integers $t < 419^5$ s.t. $t^2 ≡ 5\mod 419^5$ I'm pretty sure this has something to do with Hansel's lemma but I'm only finding examples with multi term polynomials that have been confusing ...
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Lifting point on Curve

Let $C$ be a nice (smooth, absolutely irreducible, etc...) projective algebraic curve given by $$ C: F(x,y,z)=0. $$ Suppose we have a point $(x_0:y_0:z_0)$ over $\mathbb{F}_p$ for a prime $p$ of good ...
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Intuition for geometric definition of Henselian (local) scheme?

An excerpt from section 2.3 of Néron Models: Let $R$ be a local ring with maximal ideal $\mathfrak m$ and residue field $k$. Let $S$ be the affine (local) scheme of $R$, and let $s$ be the closed ...
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$/6 = 10 \pmod {7^2}$

I am trying to understand Hensel's lemma[wiki] with Newton iteration. But I dont understand how $r_{k+1}=r_k+t p^k=r_k-\frac{f(r_k)}{f'(r_k)}$ is the same. More detailed in examples: They get $10^2 \...
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Using Hensel's Lemma to find the number of solutions.

I am trying to determine the number of solutions of the congruence $x^2 ≡ 1 \mod 2^k$ when $k \ge 3$ The statement of Hensel's Lemma that I use is the following: Attempt: $x \equiv 1,3,5,7 \mod 8$, ...
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Elementary proof for: If x is a quadratic residue mod p, then it is a quadratic residue mod p^k

In article solving quadratic congruences, it is shown how to use Hensel's lemma to iteratively construct solutions to to $x^2 \equiv a \pmod{p^k}$ from the solutions to $x^2 \equiv a \pmod{p}$. The ...
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Prove that the field of Puiseux series over $\mathbb C$ is algebraically closed

Denote by $K=\mathbb{C}((z))$ the fraction field of $\mathbb{C}[[z]]$. Define an embedding of $K$ onto itself taking $a(z)$ to $a(z^n)$ $\forall n$. The target is $\mathbb{C}((z^{1/n}))$. Define the ...
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Give a sufficient condition for the general monic quadratic polynomial $f(X)=X^2+bX+c∈\mathbb Z[X]$ to have solutions in $\mathbb{Z}/p^n\mathbb{Z}$

Let $p$ be a prime number. Combine Hensel’s lemma and quadratic reciprocity to give a sufficient condition for the general monic quadratic polynomial $$f(X) = X^2 +bX +c ∈\mathbb Z[X]$$ with integer ...
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Proving Hensel's Lemma for the ring of formal power series over the complex numbers

Suppose I have the following problem Where $A=\mathbb{C}[[z]]$ and $I^n=z^nA$, and $S_n=A/I^n$. $f$ is a monic polynomial of $\deg(f)=d$ and furthermore $\bar{f}=ev_*(f)$ is a product of polynomials $...
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Using Hensel's Lifting Lemma to Solve $x^2 + x + 34 \equiv 0 \pmod{81}$

As in the title, I'm trying to solve $$x^2 + x + 34 \equiv 0 \pmod{81}.$$ Let $f(x) = x^2 + x + 34$ throughout. I'm using Hensel's lemma, but it's a bit dense and I'm not sure my interpretation is ...
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Modular invertibility of a matrix

Consider $\mathbf{B}$ be a matrix invertable modulo a prime number $p$. Is it always possible to say that $\mathbf{B}$ is always invertable modulo $p^\alpha,\, \alpha \in \mathbb{N}$. It seems that ...
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What exactly is Hensel doing for us in this result?

I'm reading a paper where the author appeals to Hensel's lemma, but it is not clear to me quite how it is meant to be applied (or, for that matter, which version!). My commutative algebra background ...
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Hensel lifting when not a power of a prime

Say you have the equation $x^2 + x + 47 = 0$ and that you want to determine the solutions in $\mathbb{Z}/1715 \mathbb{Z}$. Note that $1715 = 7^3 \cdot 5$. Then, using Hensel's lemma, one can find the ...
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Solving equations in $\mathbb{Z}_3$ with Hensel's Lemma

Further to the post here, I'm trying to find the $n \in \mathbb{Z}$ such that there is a solution to the equation $$ x^3 +3x+y^3+3y=n$$ in $\mathbb{Z}_3$. Now, I've been able to show that in the ...
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Hensel's lemma & $p$-adic polynomial roots

I want to determine the number of roots of $f(X) = X^3-5X+20$ in $\mathbb{Z}_p$ using Hensel's lemma (lemma is on the bottom). Unfortunately I am not very well trained to solve this. Take for example $...
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Hensel's lemma modular arithmatic example problem

In an example for Hensel's Lemma we have met the criteria to use Hensel's lemma and have begun to apply it in a Hensel's iteration. We have $f(x)=x^2+1$ and our initial $x_0=2$ is a solution $\pmod{5}...
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Hensel’s Lemma Number Theory Confusion

I have been given an example, finding the solutions of the congruence $f(x) ≡ 0$ (mod $5^4$) for $f(x)=x^2+1$ This solution finds that for mod $5$ we have $x_0=2$ . So through the 'lifting' process, ...
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How do I prove this statement about $n^\text{th}$-power residues?

I am studying A Classical Introduction to Modern Number Theory by Ireland and Rosen, and the authors leave the proof of the following proposition (4.2.2) as "an exercise" ... Suppose that $a$ is ...