Hensel's Lemma/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.

Let $\map {F'} X$ be the (formal) derivative of $F$.


Let $p\Z_p$ denote the principal ideal of $\Z_p$ generated by $p$.

For all $x,y \in \Z_p$, let:

$x \equiv y \pmod {p\Z_p}$

denote congruence modulo the principal ideal $p\Z_p$.


Let $\alpha_0 \in \Z_p$ be a $p$-adic integer:

$\map F {\alpha_0} \equiv 0 \pmod {p\Z_p}$
$\map {F'} {\alpha_0} \not\equiv 0 \pmod {p\Z_p}$


Then there exists a unique $\alpha \in \Z_p$:

$\alpha \equiv \alpha_0 \pmod {p\Z_p}$
$\map F {\alpha} = 0$


Proof

Lemma 1

There exists a unique $p$-adic expansion $\ds \sum_{n = 0}^\infty d_n p^n$:
$\forall k : a_k = \ds \sum_{n = 0}^k d_n p^n$ satisfies:
$(1) \quad \map F {a_k} \equiv 0 \pmod {p^{k+1}\Z_p}$
$(2) \quad a_k \equiv \alpha_0 \pmod {p\Z_p}$

$\Box$


Let:

$\alpha = \ds \sum_{n = 0}^\infty d_n p^n$


Lemma 2

$\alpha \equiv \alpha_0 \pmod {p\Z_p}$

$\Box$


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

$\map F \alpha = 0$

$\blacksquare$

Sources