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:
 * $\map F a = 0$


 * $\forall n \in \N_{>0} : \exists a_k \in \Z : \map F {a_k} \equiv 0 \mod {p^k}$

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:
 * $a_n = \ds\sum_{j=0}^{n-1} d_j p^j$