Hensel's Lemma/P-adic Integers/Lemma 8

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 $k \in \N_{>0}$.

Then:

$x, y \in \Z_p \implies \map F {x + y p ^k} \equiv \map F x + y p^k \map {F'} x \pmod {p^{k+1}\Z_p}$

Let $\map F X = \ds \sum_{j \mathop = 0}^r c_j X^j$ where $X$ is the indeterminate and $c_0, c_1, \ldots, c_r \in \Z_p$.

Then:

$\map {F'} X = \ds \sum_{j \mathop = 1}^r j c_j X^j$

We have:

$\ds \map F {x + yp^k}$

$=$

$\ds \sum_{j \mathop = 0}^r c_j \paren{x + yp^k}^j$

Definition of $\map F X$

$\ds $

$=$

$\ds c_0 + \sum_{j \mathop = 1}^r c_j \paren{x^j + jx^{j-1}yp^k + \text{terms divisible by } p^{k+1} }$

$\ds $

$=$

$\ds c_0 + \sum_{j \mathop = 1}^r c_j x^j + \sum_{j \mathop = 1}^r j c_j x^{j-1}yp^k + \text{terms divisible by } p^{k+1}$

Rearrange terms

$\ds $

$=$

$\ds \sum_{j \mathop = 0}^r c_j x^j + \sum_{j \mathop = 1}^r j c_j x^{j-1}yp^k + \text{terms divisible by } p^{k+1}$

Combine first term with first sum

$\ds $

$=$

$\ds \sum_{j \mathop = 0}^r c_j x^j + yp^k \sum_{j \mathop = 1}^r j c_j x^{j-1} + \text{terms divisible by } p^{k+1}$

Extract common factor $yp^k$ from second sum

$\ds $

$=$

$\ds \map F x + y p^k \map {F'} x + \text{terms divisible by } p^{k+1}$

Definitions of $\map F X$ and $\map {F'} X$

$\ds $

$\equiv$

$\ds \map F x + y p^k \map {F'} a \pmod{ p^{k+1}\Z_p}$

$\blacksquare$