Roots of unity in the p-adic numbers ($\mu ({\mathbb{Q}}_p)={\mu }_{p-1}(p>2),\mu ({\mathbb{Q}}_2)={\mu }_2$)

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.