Total ramification in a prime-power cyclotomic field
= Total ramification in a prime-power cyclotomic field
{title2=$(\ell)=(1-\zeta_{\ell^r})^{\varphi(\ell^r)}$}
The prime $\ell$ is totally ramified in $\mathbb Q(\zeta_{\ell^r})$, whose degree is $\varphi(\ell^r)$. The displayed equality is an equality of ideals in its <ring of integers of a number field>, and the residue field is $\mathbb F_\ell$.