P-adic Expansion of P-adic Unit

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Theorem

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

Let $\Z_p$ be the $p$-adic integers.

Let $a \in \Z_p$.

Let $\ldots a_n \ldots a_3 a_2 a_1 a_0$ be the canonical expansion of $a$.


Then:

$a$ is a $p$-adic unit if and only if $a_0 \ne 0$

Proof

From P-adic Unit has Norm Equal to One:

a is a $p$-adic unit if and only if $\norm a_p = 1 = p^0$


By definition of the canonical expansion:

$a$ is the limit of the $p$-adic expansion $\ds \sum_{n \mathop = 0}^\infty a_n p^n$


From P-adic Norm of P-adic Expansion is determined by First Nonzero Coefficient:

$\norm a_p = p^0$ if and only if $a_0 \ne 0$


The result follows.

$\blacksquare$


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