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:
- $(1) \quad \map F {a_k} \equiv 0 \pmod {p^{k+1}\Z_p}$
- $(2) \quad a_k \equiv \alpha_0 \pmod {p\Z_p}$
- $\forall k : a_k = \ds \sum_{n = 0}^k d_n p^n$ satisfies:
$\Box$
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
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 3.4$ Hensel's Lemma, Theorem $3.4.1$
- 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.7$ Hensel's Lemma and Congruences: Theorem $1.39$