Category:Characterization of Polynomial has Root in P-adic Integers

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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:

$\map F a = 0$

if and only if:

there exists a sequence $\sequence {a_n}$ of integers:

$(1): \quad \ds \lim_{n \mathop \to \infty} {a_n} = a$

$(2): \quad \map F {a_n} \equiv 0 \mod {p^{n + 1} \Z_p}$