Definition:Canonical P-adic Expansion

Definition

Let $\struct {\Q_p, \norm {\,\cdot\,}_p}$ be the $p$-adic numbers for prime number $p$.

Let $a$ be a $p$-adic number, that is left coset, in $\Q_p$.

Let $\ds \sum_{n \mathop = m}^\infty d_n p^n$ be the unique $p$-adic expansion that is a representative of $a$ where:

$m \in \Z_{\le 0}$

$\forall n \in \Z_{\ge m}: d_n$ is a $p$-adic digit

$m < 0 \implies d_m \ne 0$

For $m < 0$, the canonical $p$-adic expansion of $a$ is the expression:

$\ldots d_n \ldots d_3 d_2 d_1 d_0 . d_{-1} d_{-2} \ldots d_m$

with (countably) infinitely many $p$-adic digits before the point and finitely many $p$-adic digits after the point.

In the case that $m = 0$, the canonical $p$-adic expansion of $a$ is the expression:

$\ldots d_n \ldots d_3 d_2 d_1 d_0$

with (countably) infinitely many $p$-adic digits to the left and no point or $p$-adic digits after the point.

Also known as

The canonical $p$-adic expansion of $a$ is simply called the canonical expansion of $a$ when $a$ is understood to be a $p$-adic number.

Also see

• P-adic Number has Unique P-adic Expansion Representative

• P-adic Number is Limit of Unique P-adic Expansion

• P-adic Expansion Converges to P-adic Number iff P-adic Expansion Represents P-adic Number

Sources

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