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 $\alpha_0 \in \Z_p$ be a $p$-adic integer:
 * $\map F {\alpha_0} \equiv 0 \pmod {p\Z_p}$
 * $\map {F'} {\alpha_0} \not\equiv 0 \pmod {p\Z_p}$

Then there exists a unique $\alpha \in \Z_p$:
 * $\alpha \equiv \alpha_0 \pmod {p\Z_p}$
 * $\map F {\alpha} = 0$

Lemma 1
Let:
 * $\alpha = \ds \sum_{n = 0}^\infty d_n p^n$

Lemma

 * $\alpha \equiv \alpha_0 \pmod {p\Z_p}$

Lemma

 * $\map F {\alpha} = 0$