Questions tagged [hensels-lemma]
For questions regarding Hensel's lifting lemma in modular arithmetic and its generalization to commutative rings.
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Enumerating solutions of $x^2+y^2\equiv 1 \mod p^k$ with Hensel's lemma
I'd like to count the number of solutions to the equation
$$x^2 +y^2\equiv 1 \bmod p^k.$$
The case of $k=1$ is discussed here along with the number of solutions in a field of $p^k$ elements, so using ...
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Example of a complete valued field that does not satisfy hensel's lemma
A valued field $(K,v)$ is henselian if it satisfies hensel's lemma.
Notation : $O_v$ is the valuation ring of $(K,v)$ (it is a local ring), $F_v$ is the residue field of $F_v$.
Hensel's Lemma : for ...
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Uniqueness in Hensel's lemma
Henri Cohen's "A Course in Computational Algebraic Number Theory contains the following variant of Hensel's lemma.
Theorem 3.5.3. Let $p$ be a prime, and let $C,A_e,B_e,U,V$ be polynomials with ...
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What's a direct or quickest proof of simple zeros lift to factorizations lift?
Is there a direct proof that for a valued field $(F,v)$, the existence of lifts of simple roots in $k_v$ (the residue field) to $\mathcal{O}_v$ (the valuation subring) implies that factorizations in $...
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A generalization of Hensel lemma
Let A be a noetherian ring, let I be a proper
ideal of A. Suppose that A is complete and separate for the I-adic topology. Let
F(X) ∈ A[X] be a polynomial such that there exists a ∈ A with F(a) ∈ I ...
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Analytic parameterization of a residue disk
Let $C/\mathbb{Z}_p$ be a non-singular, projective, curve, with good reduction. Now, let $P\in C(\mathbb{F}_p)$ be a point of its special fibre, and denote by $\mathcal{D}$ the residue disk of $C$, ...
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Number of solutions to system of polynomial equations $\!\bmod p^k$
Consider the system of polynomial equations
$$
\begin{cases}
a(X^2+Y^2)^2-2b(X^2+Y^2)+4(bX-cY)X \equiv 0 &\pmod{p^k} \\
d(X^2+Y^2)^2+2c(X^2+Y^2)+4(bX-cY)Y \equiv 0 &\pmod{p^k}
\end{cases}
$$
...
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How often is a monic polynomial highly divisible by p?
Let $p$ be a prime, and let $|\cdot|_p$ be the $p$-adic absolute value. Let $f(x) \in \mathbb{Z}[x]$ be a monic polynomial with $f(0) = 1$.
Question 1 What is the volume $c_n$ of the following set
$$\{...
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Connection between roots in $Q_p$ and $Z_p$
I'm working on some problems where I have to find solutions in $Z_p$ and $Q_p$ of polynomials of the form $ax^2+by^2=1$. I've seen Hensel's lemma for solution over $Z_p$. For solutions over $Q_p$. I'...
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Find prime number which satisfies $p \in {\Bbb{{Q}_2}^{\times}}^2-{{\Bbb{Q}_2}^{\times}}^4$
What is an minimal prime number $p$ which satisfies $p \in {\Bbb{{Q}_2}^{\times}}^2-{{\Bbb{Q}_2}^{\times}}^4$?
For $a \in \Bbb{Z}$, if $a≡1,3,4,7$mod8, then for each $a$, $\Bbb{Q}_2$ is $\Bbb{Q}_2(\...
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Let $K$ be a number field and $v$ be its place. Let $K_v$ be completion of $K$ at $v$. I want to prove $y^2=x^4-p$ has $K_v$ rational point.
Let $K$ be a number field and $v$ be its place. Let $p$ be a prime element of ring of integers of $K$. Let $K_v$ be completion of $K$ at $v$.
I want to prove $C: y^2=x^4-p$ has $K_v$ rational point.
I ...
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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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Applying Hensel's lemma to solve $x^2 + 8 \equiv 0\pmod {121}$. [duplicate]
When solving for $x^2 + 8 \equiv 0 \pmod {121}$, How can we apply Hensel's lemma to solve for its solutions? What I currently understand is that for a prime $p$ and $e \geq 2$, then $f(x) \equiv 0 \...
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$\underset{\bar{a},\bar{b}\in S^n}{Pr}[f(x,\bar{a}t+\bar{b})\text{ is reducible and }f(x,\bar{b})\text{ is square-free}]\leq\frac{7d^6}{|S|}$
Let $\mathbb{F}$ be a field. Let $S\subset \mathbb{F}$ be a finite set
with a size large enough. Let
$f(x,y_1,y_2,\dots,y_n)=f(x,\overline{y})$ be a almost monic and
irreducible polynomial in total ...
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Henselisation of Normal local rings (in Milne's Etale Cohomology)
The usual way to define the Henselisation $A^h$ of a local ring $(A, \mathfrak{m})$ is to take direct limit $\varinjlim (B, q)$ over all etale neighborhoods of $A$
(i.e. pairs $(B,q)$ where $B$ is an ...
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