Total ramification in a prime-power cyclotomic field ($(\ell )=(1-{\zeta }_{{\ell }^r}{)}^{\phi ({\ell }^r)}$)

The prime $\ell$ is totally ramified in $\mathbb{Q}({\zeta }_{{\ell }^r})$, whose degree is $\phi ({\ell }^r)$. The displayed equality is an equality of ideals in its ring of integers of a number field, and the residue field is ${\mathbb{F}}_{\ell }$.