Roots of unity in a p-adic field ($\mu (K)$)

The roots of unity in a finite extension $K/{\mathbb{Q}}_p$ form a finite group. Each element of their prime-to-$p$ part is a Teichmuller representative, while any $p$-power part lies among the principal units.

In a mixed-characteristic discretely valued field, a primitive $p^a$th root of unity reduces to one. For every $j$ prime to $p$, the ratio $({\zeta }^j-1)/(\zeta -1)$ is a unit, so all factors of ${\Phi }_{p^a}(1)=p$ have the same valuation. Taking valuations gives the formula. Because the left side is a positive integer for a normalized valuation, the existence of such a root forces $p^{a-1}(p-1)\mid v(p)$, bounding the possible $p$-power torsion.

For odd $p$, both possibilities occur at $v(p)=p-1$: ${\mathbb{Q}}_p({\zeta }_p)$ contains $p$-torsion, while ${\mathbb{Q}}_p(\sqrt[p-1]{p})$ does not. Both extensions are totally ramified of degree $p-1$ by the Eisenstein criterion. In the second field, if ${\zeta }_p=1+u\pi$ with ${\pi }^{p-1}=p$, the cyclotomic equation reduced after division by $p$ would give $1+{\bar{u}}^{p-1}=0$ in ${\mathbb{F}}_p$, contradicting ${\bar{u}}^{p-1}=1$. For $p=2$, $v(2)=1$ always permits the root $-1$, so the torsion exponent is exactly one and the zero-exponent alternative is impossible.

If a finite extension of the P-adic numbers contains a primitive $p$th root of unity $\zeta$, the identity $p={\prod }_{j=1}^{p-1}(1-{\zeta }^j)$ gives $e=(p-1)v_L(1-\zeta )\ge p-1$. The ratios of these factors reduce to the nonzero integers $j$ in the residue field. Thus an extension with $e<p-1$ has only roots of unity of order prime to $p$.

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.

The Eisenstein polynomial of $1-i$ is $T^2-2T+2$, so ${\mathbb{Q}}_2(i)$ has residue field ${\mathbb{F}}_2$ and ramification index two. The four roots $1,i,-1,-i$ have distinct cosets modulo $U_3$, which has index four by the successive quotients of principal-unit groups. The logarithm isomorphism on deep principal units gives $U_3\cong ({\mathbb{Z}}_2[i],+)$. Hence these four roots split the full unit group, and are all its torsion.

For $K={\mathbb{Q}}_3({\zeta }_3)$, the element $\pi ={\zeta }_3-1$ is a uniformizer and