Characterization of Polynomial has Root in P-adic Integers/Necessary Condition

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 $a \in \Z_p$.

Let $\map F a = 0$.

Then:

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 $p^{n + 1} \Z_p$

Let $\map F a = 0$.

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

For all $n \in \N_{>0}$, let:

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

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

$\forall n \in \N_{>0} : a_n \in \Z$

We have:

$\ds \forall n \in \N_{>0}: \, $

$\ds a_n$

$\equiv$

$\ds a \pmod {p^n \Z_p}$

$\ds \leadsto \ \ $

$\ds \forall n \in \N_{>0}: \, $

$\ds \map F {a_n}$

$\equiv$

$\ds \map F a \pmod {p^n \Z_p}$

$\ds $

$\equiv$

$\ds 0 \pmod {p^n\Z_p}$

as $\map F a = 0$

$\ds \leadsto \ \ $

$\ds \forall n \in \N_{>0}: \, $

$\ds \map F {a_n}$

$\equiv$

$\ds 0 \pmod {p^n}$

$\blacksquare$