User:Barto/Hensel's Lemma/First Form

Theorem
Let $p$ be a prime number.

Let $k>0$ be a positive integer.

Let $f(X) \in \Z[X]$ be a polynomial.

Let $x_k\in\Z$ such that:
 * $f(x_k)\equiv 0 \pmod{p^k}$.
 * $f'(x_k)\not\equiv 0 \pmod{p}$.

Then for every integer $l\geq 0$, there exists an integer $x_{k+l}$ such that:
 * $f(x_{k+l})\equiv 0 \pmod{p^{k+l}}$
 * $x_{k+l}\equiv x_k\pmod{p^k}$

and any two integers satisfying these congruences are congruent modulo $p^{k+l}$.

Moreover, for all $l\geq0$ and any solutions $x_{k+l}$ and $x_{k+l+1}$:
 * $x_{k+l+1}\equiv x_{k+l}-\frac{f(x_{k+l})}{f'(x_{k+l})}\pmod{p^{k+l+1}}$
 * $x_{k+l+1}\equiv x_{k+l}\pmod{p^{k+l}}$

Proof
By induction. For $l=0$ this is obvious.

Let $x_{k+l}$ be as stated.

We show that there exists a unique lift $x_{k+l+1}$ of $x_{k+l}$ up to a multiple of $p^{k+l+1}$.

Let $x_{k+l+1}=x_{k+l}+p^{k+l}t$ be such a lift.

Then $f(x_{k+l+1})\equiv f(x_{k+l}) +p^{k+l}t f'(x_{k+l})$, where $f'(x_{k+l})\equiv f'(x_k)\not\equiv0\pmod p$.

Then $f(x_{k+l+1})\equiv0 \pmod{p^{k+l+1}}$ $t\equiv-\frac{f(x_{k+l})}{p^{k+l}f'(x_{k+l})}\pmod p$, which proves that there is a unique solution modulo $p^{k+l+1}$.