# Questions tagged [hensels-lemma]

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

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### Ring isomorphism via Hensel lifting

Consider a monic polynomial $f\in\mathbb Z[X]$ (see Note). Assume that we can factor $f\bmod p=gh$ into two monic irreducible polynomials $g\neq h\in\mathbb F_p[X]$ of same degree (or more generally, ...
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### 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 $R$ ...
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### 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 ...
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### 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,...
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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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