# Questions tagged [hensels-lemma]

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

57 questions
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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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### 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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### 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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### 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 ...
1answer
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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 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, ...
1answer
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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 ...