Hensel's Lemma/P-adic Integers/Lemma 9

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

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

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

Let $T$ be the set of $p$-adic digits.

For each $k \in \N_{>0}$, let:

$S_k = \set{\tuple{b_0, b_1, \ldots, b_{k-1}} \subseteq T^k : \map F {\ds \sum_{n = 0}^{k-1} b_n p^n} \equiv 0 \pmod{p^k\Z_p} \quad \text{and} \quad \ds \sum_{n = 0}^{k-1} b_n p^n \equiv \alpha_0 \pmod{p\Z_p}}$

Let:

$\tuple{b_0, b_1, \ldots, b_{k-1}} \in S_k$.

Then:

$\paren{c, d \in T : \tuple{b_0, b_1, \ldots, b_{k-1}, c}, \tuple{b_0, b_1, \ldots, b_{k-1}, d} \in S_{k+1}} \implies c = d$

Proof

Lemma 7

$x \in \Z_p \implies \exists y \in T : y p^k \equiv x p^k \pmod {p^{k+1}\Z_p}$

$\Box$

Lemma 8

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

$\Box$

Lemma 10

$\forall x \in \Z_p: p^k x \equiv 0 \pmod{p^{k+1}\Z_p} \implies x \equiv 0 \pmod{p\Z_p}$

$\Box$

Let:

$c, d \in T : \tuple{b_0, b_1, \ldots, b_{k-1}, c}, \tuple{b_0, b_1, \ldots, b_{k-1}, d} \in S_{k+1}$

Let:

$a = \ds \sum_{n = 0}^{k-1} b_n p^n$

We have:

\(\ds a\)

\(\equiv\)

\(\ds \alpha_0 \pmod{p\Z_p}\)

Hypothesis

\(\ds \leadsto \ \ \)

\(\ds \map {F'} a\)

\(\equiv\)

\(\ds \map {F'} {\alpha_0} \pmod{p\Z_p}\)

Polynomials of Congruent Ring Elements are Congruent

From Lemma 7:

$\exists b \in T : \map {F'} a \equiv b \pmod{p\Z_p}$

We have:

\(\ds b\)

\(\equiv\)

\(\ds \map {F'} a \pmod{p\Z_p}\)

From above

\(\ds \)

\(\equiv\)

\(\ds \map {F'} {\alpha_0} \pmod{p\Z_p}\)

From above

\(\ds \)

\(\not\equiv\)

\(\ds 0 \pmod{p\Z_p}\)

Hypothesis

\(\ds \leadsto \ \ \)

\(\ds p\)

\(\nmid\)

\(\ds b\)

We have:

\(\ds 0\)

\(\equiv\)

\(\ds \map F {a + cp^k} - \map F {a + dp^k} \pmod{p^{k+1}\Z_p}\)

By hypothesis, $\map F {a + cp^k} \equiv \map F {a + dp^k} \equiv 0 \pmod{p^{k+1}\Z_p}$

\(\ds \)

\(\equiv\)

\(\ds \map F a + cp^k \map {F'} a - \map F a - dp^k \map {F'} a \pmod{p^{k+1}\Z_p}\)

Lemma 8

\(\ds \)

\(\equiv\)

\(\ds cp^k \map {F'} a - dp^k \map {F'} a \pmod{p^{k+1}\Z_p}\)

$\map F a$ terms cancel

\(\ds \)

\(\equiv\)

\(\ds p^k \map {F'} a \paren{c - d} \pmod{p^{k+1}\Z_p}\)

Extract common factor $p^k\map {F'} a$

\(\ds \leadsto \ \ \)

\(\ds 0\)

\(\equiv\)

\(\ds \map {F'} a \paren{c - d} \pmod{p\Z_p}\)

Lemma 10

\(\ds \)

\(\equiv\)

\(\ds b \paren{c - d} \pmod{p\Z_p}\)

As $b \equiv \map {F'} a \pmod{p\Z_p}$

\(\ds \leadsto \ \ \)

\(\ds p\)

\(\divides\)

\(\ds b \paren{c - d}\)

Congruence Modulo Equivalence for Integers in P-adic Integers

\(\ds \leadsto \ \ \)

\(\ds p\)

\(\divides\)

\(\ds c - d\)

Euclid's Lemma for Prime Divisors since $p \nmid b$

\(\ds \leadsto \ \ \)

\(\ds c\)

\(\equiv\)

\(\ds d \pmod p\)

Definition of Congruence Modulo Integer

\(\ds \leadsto \ \ \)

\(\ds c\)

\(=\)

\(\ds d\)

As $0 \le x, y < p$

The result follows.

$\blacksquare$