P-adic Numbers are Uncountable

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Theorem

Let $p$ be any prime number.

The set of $p$-adic numbers $\Q_p$ is an uncountable set.

Proof

Let $P$ be the set of sequences on $\set{i : i \in \N : 0 \le i < p}$.

That is:

$P = \set{\sequence{d_n} : d_n \in \N : 0 \le d_n < p}$

From Cantor's Diagonal Argument:

$P$ is an uncountable set

Let $f: P \to \Q_p$ be the mapping from $P$ to $\Z_p$ defined by:

$\forall \sequence{d_n} \in P : \map f {\sequence{d_n}} = \ds \sum_{n = 0}^\infty d_n p^n$

where $Z_p$ denotes the $p$-adic integers and $\ds \sum_{n = 0}^\infty d_n p^n$ denotes a $p$-adic expansion

From P-adic Integer has Unique P-adic Expansion Representative:

$f$ is bijective

Hence:

$\Z_p$ is an uncountable set

Recall that $\Z_p \subseteq \Q_p$.

From Sufficient Conditions for Uncountability:

$\Q_p$ is an uncountable set

$\blacksquare$

Sources

• 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.4$ The field of $p$-adic numbers $\Q_p$: Exercise $20$