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Questions tagged [hensels-lemma]

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

Hensel's Lemma. If a polynomial equation has a simple root modulo a prime number $p$, then this root corresponds to a unique root of the same equation modulo any higher power of $p$, which can be found by iteratively "lifting" the solution modulo successive powers of $p$.

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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 point of $S$. From a geometric point of view,…
Arrow
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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 a $p$-adic field $\Bbb{Q}_p$. (With a little…
Torsten Schoeneberg
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Coding Theory and Algebra

Why is Algebra so useful in Coding Theory? I know a little bit of algebra and I just know what codes are. I appreciate it if someone can give a brief explanation of how/ in what sense is algebra useful in Coding Theory. More specifically, I recently…
user59756
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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 that we might have $f'(a_n) \equiv 0 \pmod p$ so…
Alphonse
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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 lemma (please tell me when it is already…
TP.TPPR
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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 case where $p=2$ is treated separately. While the…
Sadeq Dousti
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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 that $\mathbb{Z}/p\mathbb{Z}$ contains a primitive…
Oromis
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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 inseparable (and irreducible) polynomial. Let $\mathcal{P}$…
BharatRam
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Hensel lifting square roots $\!\bmod p\,$ to $\!\bmod p^2$

I've been working on this problem for a while, but hit a dead end. Here's the problem: Suppose $p$ is an odd prime. Also let $b^2 \equiv a \pmod p$ and $p$ does not divide $a$. Prove there exists some $k \in \mathbb{Z}$ such that $(b+kp)^2 \equiv a…
MathNewbie
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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 is unfortunately not as good as I'd like, and I…
theHigherGeometer
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Understanding Hensel's Lemma

I am learning Hensel's Lemma and trying to solve the polynomial congruence $$x^5+x^4+1\equiv 0\pmod{81}$$ Now my professor taught us the technique of building up from $p$, to $p^2$, and continuing to $p^k$ by repeated use of the lemma. I think this…
Lalaloopsy
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Hensel's Lemma vs "Hensel Lifting"

I am reading the textbook "Finite Fields and Galois Rings" by Wan, and am confused by the definition of a Hensel lift and how it relates to Hensel's Lemma. The result that is labelled Hensel's Lemma (Lemma 13.7) is the existence, for any…
popstack
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If $y$ is a $k$th power modulo $p^\gamma$, then it is also a $k$th power modulo $p^t$ for $t \geqslant \gamma$

This question is the true version I wanted to ask of this question. Say $p$ is an odd prime number, $k$ a positive integer and $p^{\tau} || k$. Let $\gamma = \tau + 1$. I would like to prove If $y \in \mathbf{Z}$ is a $k$-th power modulo…
Gory
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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 $x(t) \in k[[t]]$ with $x(0) = a$ and $f(t,x(t)) = 0$…
54321user
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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 has that the map $X(R) \to X(R/I)$ is bijective. One…
Asvin
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