Roots of unity in the p-adic numbers
= Roots of unity in the p-adic numbers
{title2=$\mu(\mathbb Q_p)=\mu_{p-1}\ (p>2),\quad \mu(\mathbb Q_2)=\mu_2$}
For odd $p$, <Teichmuller representatives> provide the $p-1$ prime-to-$p$ roots, while the <logarithm isomorphism on deep principal units> makes $1+p\mathbb Z_p$ torsion-free. At two, the decomposition $\mathbb Z_2^\times=\{1,-1\}\times(1+4\mathbb Z_2)$ leaves exactly two roots.