= Total ramification criterion for local cyclotomic extensions
{title2=$\mathbb Q_p(\zeta_n)/\mathbb Q_p\text{ is totally ramified}\iff n=p^a m,\ m\mid p-1$}
Here $a\geq0$ 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 $p\equiv1\pmod m$.
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