Questions tagged [hensels-lemma]

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

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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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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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Equivalence of generalized Hensel's lemmas

Let $(A,\mathfrak{p})$ be a local ring complete in the The usual version of Hensel's lemma states that if $f(a)\equiv 0\bmod{\mathfrak{p}}$ and $f'(a)\not\equiv 0\bmod{\mathfrak{p}}$, then there is ...
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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 $r$ ...
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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 $a$...
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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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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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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 ...
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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; $A^\text{h}$...