Characterization of Polynomial has Root in P-adic Integers

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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:

$\map F a = 0$

if and only if:

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


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


Proof

Necessary Condition

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:

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


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:

\(\ds \forall n \in \N_{>0}: \, \) \(\ds a_n\) \(\equiv\) \(\ds a \pmod {p^n \Z_p}\) Partial Sum Congruent to P-adic Integer Modulo Power of p
\(\ds \leadsto \ \ \) \(\ds \forall n \in \N_{>0}: \, \) \(\ds \map F {a_n}\) \(\equiv\) \(\ds \map F a \pmod {p^n \Z_p}\) Polynomials of Congruent Ring Elements are Congruent
\(\ds \) \(\equiv\) \(\ds 0 \pmod {p^n\Z_p}\) as $\map F a = 0$
\(\ds \leadsto \ \ \) \(\ds \forall n \in \N_{>0}: \, \) \(\ds \map F {a_n}\) \(\equiv\) \(\ds 0 \pmod {p^n}\) Congruence Modulo Equivalence for Integers in P-adic Integers

$\Box$


Sufficient Condition

Let there exist 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:

\(\ds \forall k \in \N: \, \) \(\ds a\) \(\equiv\) \(\ds a_k \pmod {p^{n + 1} \Z_p}\) Partial Sum Congruent to P-adic Integer Modulo Power of p
\(\ds \leadsto \ \ \) \(\ds \forall k \in \N: \, \) \(\ds \map F a\) \(\equiv\) \(\ds \map F {a_k} \pmod {p^{n + 1} \Z_p}\) Polynomials of Congruent Ring Elements are Congruent
\(\ds \) \(\equiv\) \(\ds 0 \pmod {p^{n + 1} \Z_p}\) as $\map F {a_k} \equiv 0 \pmod {p^{n + 1} \Z_p}$
\(\ds \leadsto \ \ \) \(\ds \forall k \in \N: \, \) \(\ds \map F a\) \(\in\) \(\ds p^{n + 1} \Z_p\) Definition of Congruence Modulo an Ideal
\(\ds \leadsto \ \ \) \(\ds \forall k \in \N: \, \) \(\ds \norm {\map F a}_p\) \(\le\) \(\ds p^{-k - 1}\) Characterization of Closed Ball in P-adic Numbers
\(\ds \leadsto \ \ \) \(\ds \norm {\map F a}_p\) \(\le\) \(\ds \lim_{k \mathop \to \infty} p^{-k - 1}\) Squeeze Theorem for Real Sequences
\(\ds \) \(=\) \(\ds 0\) Sequence of Powers of Number less than One
\(\ds \leadsto \ \ \) \(\ds \norm {\map F a}_p\) \(=\) \(\ds 0\) $p$-adic norm is a mapping from $\Q_p$ to the non-negative reals
\(\ds \leadsto \ \ \) \(\ds \map F a\) \(=\) \(\ds 0\) Non-Archimedean Norm Axiom $\text N 1$: Positive Definiteness

$\blacksquare$

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