P-adic Norm not Complete on Rational Numbers/Proof 2

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Theorem

Let $\norm {\,\cdot\,}_p$ be the $p$-adic norm on the rationals $\Q$ for some prime $p$.

Then:

the valued field $\struct {\Q, \norm {\,\cdot\,}_p}$ is not complete.


That is, there exists a Cauchy sequence in $\struct {\Q, \norm{\,\cdot\,}_p}$ which does not converge to a limit in $\Q$.


Proof

Hensel's Lemma is used to prove the existence of a Cauchy sequence that does not converge.


Lemma 1

$\exists x \in \Z_{>0}: p \nmid x, x \ge \dfrac {p + 1} 2$

$\Box$


Let $x_1 \in \Z_{>0}: p \nmid x_1, x_1 \ge \dfrac {p + 1} 2$


Let $k$ be a positive integer such that $k \ge 2, p \nmid k$.


Let $a = x_1^k + p$.


Lemma 2

$a \in \Z_{> 0}: \nexists \,c \in \Z : c^k = a$

$\Box$


Let $\map f X \in \Z \sqbrk X$ be the polynomial:

$X^k - a$


Lemma 3

$\map f {x_1} \equiv 0 \pmod p$

$\Box$


Let $\map {f'} X \in \Z \sqbrk X$ be the formal derivative of $\map f X$.


Lemma 4

$\map {f'} {x_1} \not \equiv 0 \pmod p$

$\Box$


From Hensel's Lemma there exists a sequence of integers $\sequence {x_n}$ such that:

$(1) \quad \forall n: \map f {x_n} \equiv 0 \pmod {p^n}$
$(2) \quad \forall n: x_{n + 1} \equiv x_n \pmod {p^n}$


Lemma 5

$\ds \lim_{n \mathop \to \infty} {x_n}^k = a$ in $\struct {\Q, \norm {\,\cdot\,}_p}$

$\Box$


From Sequence of Consecutive Integers Modulo Power of p is Cauchy in P-adic Norm then:

$\sequence {x_n}$ is a Cauchy sequence in $\struct {\Q, \norm {\,\cdot\,}_p}$


Aiming for a contradiction, suppose $\sequence {x_n}$ is a sequence such that for some $c \in \Q$:

$\ds \lim_{n \mathop \to \infty} x_n = c$

in $\struct {\Q, \norm {\,\cdot\,}_p}$

From Product Rule for Sequences in Normed Division Ring then:

$\ds \lim_{n \mathop \to \infty} x_n^k = c^k$

Hence:

$c^k = a \in \Z$.

From Nth Root of Integer is Integer or Irrational then:

$c \in \Z$

This contradicts Lemma 2.

So the sequence $\sequence {x_n}$ does not converge in $\struct {\Q, \norm{\,\cdot\,}_p}$.

The result follows.

$\blacksquare$


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