# Questions tagged [hensels-lemma]

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

79 questions
Filter by
Sorted by
Tagged with
27 views

117 views

39 views

### Hensel's Lemma - Unique in what sense?

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

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 ...
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 ...
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 ...
91 views

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 ...
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 ...
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 ...
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. ...
38 views

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 ...
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 ...
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"?
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 $x(t) \in k[[t]]$...
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 ...