User:Leigh.Samphier/Sandbox/Hensel's Lemma/P-adic Integers/Corollary

Theorem
Let $\Z_p$ be the $p$-adic integers for some prime $p$.

Let $\map f X \in \Z_p \sqbrk X$ be a polynomial.

Let $\map {f'} X$ be the (formal) derivative of $f$.

Let $a_0 \in \Z_p$ be a $p$-adic integer:
 * $\norm{ \map f {a_0}}_p \le \dfrac 1 p$
 * $\norm{\map {f'} {a_0}}_p > \dfrac 1 p$

Then there exists a unique $a \in \Z_p$:
 * $\norm{a - a_0}_p \le \dfrac 1 p$
 * $\map {f'} {a} = 0$