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:
 * $\map F a = 0$


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

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

Proof
From Characterization of Polynomial has Root in P-adic Integers:
 * $\map F a = 0$


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

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

From Congruence Modulo Equivalence for Integers in P-adic Integers:
 * $\map F a = 0$


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