Lagmental Vicfred

Simple Roots Lift Uniquely by Hensel's Lemma by Vicfred

tags: mathematics, p-adic numbers, valuations, Hensel lemma

A root modulo p with nonzero derivative extends to a compatible root modulo every power of p. The example is deliberately concrete; it is a test of the statement, not a substitute for it.

Definitions first

The valuation \(v_p(n)\) counts factors of \(p\), and the metric \(|x|_p=p^{-v_p(x)}\) reverses the usual sense of size. Hensel lifting turns approximate roots modulo \(p\) into exact \(p\)-adic roots.

$$ f(a_1)\equiv0\pmod p,\qquad f'(a_1)\not\equiv0\pmod p $$

The formulas should not be merged too early. The datum \(\mathsf D\), the conclusion \(\mathsf C\), and the bridge \(\Longrightarrow\) have three different logical jobs.

$$ \exists!\,a\in\mathbf Z_p,\qquad a\equiv a_1\pmod p,\quad f(a)=0 $$

A small case in full

This is the algebraic core of the note. Once this line is correct, the surrounding interpretation has something solid to refer to.

$$ a_{n+1}=a_n-f(a_n)f'(a_n)^{-1}\pmod{p^{2^n}} $$

The reusable statement

The abstraction earns its keep by explaining why the same computation reappears. The notation compresses repeated reasoning without erasing the hypothesis that licenses it.

$$ \begin{aligned} \mathsf{D}\;&:\quad f(a_1)\equiv0\pmod p,\qquad f'(a_1)\not\equiv0\pmod p,\\[5pt] \mathsf{C}\;&:\quad \exists!\,a\in\mathbf Z_p,\qquad a\equiv a_1\pmod p,\quad f(a)=0. \end{aligned} $$

A nearby false statement

A congruence root lifts uniquely only in the simple-root case. Multiple roots need stronger inequalities and may split, disappear, or lift nonuniquely.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \exists!\,a\in\mathbf Z_p,\qquad a\equiv a_1\pmod p,\quad f(a)=0 \end{gathered}} $$

I would use the boxed line as a reference later, while returning to the full display whenever a hypothesis becomes uncertain. That division keeps compression from becoming ambiguity.

This article was posted on Thu 23 June 2011. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.