= p-adic unit
{title2=$\mathbb Z_p^\times$}
= p-adic unit group
{c}
{synonym}
A <p-adic unit> is an invertible element of the <ring> $\mathbb Z_p$. For a <p-adic integer> $x$, this is equivalent to $v_p(x)=0$, $|x|_p=1$, or a nonzero initial <p-adic digit> $b_0$. A <rational number> $a/b$ is a <p-adic unit> when neither its reduced <numerator> nor its <denominator> is divisible by $p$.
Thus $3$ and $1/3$ are <p-adic units> at $p=2$, while $2$ is a <p-adic integer> with no inverse inside $\mathbb Z_2$. The <modular inversion converges to a p-adic inverse> construction computes the inverse of an integer unit to arbitrary finite precision.
Back to article page
