Characterization of Polynomial has Root in P-adic Integers

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

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

Let $a \in \Z_p$.

Then:
 * $\map F a = 0$


 * there exists a sequence $\sequence{a_n}$ of integers:
 * $(1)\quad\ds\lim_{n \mathop \to \infty} {a_n} = a$
 * $(2)\quad\map F {a_n} \equiv 0 \mod {p^{n+1}\Z_p}$

where $\map F {a_n} \equiv 0 \mod {p^{n+1}\Z_p}$ denotes congruence modulo the ideal $p^{n+1}\Z_p$

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:
 * $\ds\lim_{n \mathop \to \infty} {a_n} = a$

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

We have:

Sufficient Condition
Let there exists a sequence $\sequence{a_n}$ of integers:
 * $(1)\quad\ds\lim_{n \mathop \to \infty} {a_n} = a$
 * $(2)\quad\map F {a_n} \equiv 0 \mod {p^{n+1}\Z_p}$

We have: