Characterization of Integer Polynomial has Root in P-adic Integers

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

Let $\map F X \in \Z \sqbrk X$ be a polynomial with integer coefficients.

Let $a \in \Z_p$.

Then:
 * $\map F a = 0$


 * $\forall n \in \N_{>0} : \exists a_n \in \Z :$
 * $(1)\quad\ds\lim_{n \mathop \to \infty} {a_n} = 0$
 * $(2)\quad\map F {a_n} \equiv 0 \mod {p^n}$

That is, a polynomial with integer coefficients has a root it has an integer root modulo $p^k$ for every $k \in \N_{>0}$.

Necessary Condition
Let $\map F a = 0$.

Let $a = \ds\sum_{j=0}^\infty d_j p^j$ be the $p$-adic expansion of $a$.

For all $n \in \N_{>0}$, let:
 * $a_n = \ds\sum_{j=0}^{n-1} d_j p^j$

By definition of $p$-adic expansion of a $p$-adic integer:
 * $\forall n \in \N_{>0} : a_n \in \Z$

By definition of a polynomial with integer coefficients:
 * $\forall n \in \N_{>0} : \map F {a_n} \in \Z$

We have: