Definition:P-adic Unit

Definition

Let $p$ be a prime number.

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

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

The set of $p$-adic units, denoted $\Z_p^\times$, is the set of invertible elements of $\Z_p$.

Also see

• P-adic Unit has Norm Equal to One where it is shown that the $p$-adic units is:

$\Z_p^\times = \set {x \in \Q_p: \norm x_p = 1}$

• P-adic Expansion of P-adic Unit where it is shown that the $p$-adic units is:

$\Z_p^\times = \set {\ds \sum_{n \mathop = 0}^\infty a_n p^n \in \Q_p: a_0 \ne 0}$

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 3.3$ Exploring $\Q_p$

• 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.5$ Arithmetical operations in $\Q_p$