Cyclotomic extension of a p-adic field

The cyclotomic extension ${\mathbb{Q}}_p({\zeta }_{p^n})/{\mathbb{Q}}_p$ is generated by a primitive $p^n$th root of unity. It is totally ramified and has Galois group $(\mathbb{Z}/p^n\mathbb{Z}{)}^{\times }$.

Here $a\ge 0$ and $p\nmid m$. The prime-to-$p$ roots of unity generate an unramified extension of degree equal to the multiplicative order of $p$ modulo $m$; for $m=1$ this degree is one. Indeed, their distinct residues lift uniquely by Hensel lemma, and the Frobenius automorphism acts by raising them to $p$. The $p$-power roots of unity generate a totally ramified extension, because the translated cyclotomic polynomial ${\Phi }_{p^a}(1+T)$ is Eisenstein. The two fields have trivial intersection and their compositum has the same residue degree as the prime-to-$p$ field, by unramified base change preserves total ramification. Thus total ramification is equivalent to $p\equiv 1(\mathrm{mod}m)$.