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

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:
 * $\map f {a_0} \equiv 0 \pmod {p\Z_p}$
 * $\map {f'} {a_0} \not\equiv 0 \pmod {p\Z_p}$

Then there exists a unique $a \in \Z_p$:
 * $a \equiv a_0 \pmod {p\Z_p}$
 * $\map {f'} {a} = 0$