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 $a$ with $p$-adic digits:
 * $\set{d_n: n \in \N, -m \le n}$

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 infinitely many $p$-adic digits before the point and finitely many $p$-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