Decomposition of an unramified prime in a cyclotomic field

Given: A cyclotomic field $K=\mathbb{Q}({\zeta }_f)$ presented by its conductor $f$, so that $f$ is the least admissible index for $K$ (Cyclotomic conductor of a full cyclotomic field), a rational prime $\mathrm{\ell }$ with $\mathrm{\ell }\nmid f$, and the factorisation $f={\mathrm{\ell }}^{a}m$ with $\gcd (\mathrm{\ell },m)=1$ (so $a=0$ and $m=f$ under the hypothesis $\mathrm{\ell }\nmid f$).

The conductor $f$ of a full cyclotomic field is a reduced index: it is odd or divisible by $4$ (Conductor of a full cyclotomic field, Cyclotomic conductor of a full cyclotomic field).

Prime factorisation in a reduced cyclotomic field: for the reduced index $f$, a rational prime $\mathrm{\ell }$, and $f={\mathrm{\ell }}^{a}m$ with $\gcd (\mathrm{\ell },m)=1$, $\mathrm{\ell }{\mathcal{O}}_K=(P_1\cdots P_g{)}^e,e=\phi ({\mathrm{\ell }}^a),d={\mathrm{ord}}_m(\mathrm{\ell }),g=\phi (m)\mathrm{/}d,$ with the $P_i$ pairwise distinct primes of residue degree $d$; here ${\mathrm{ord}}_1(\mathrm{\ell })=1$ (Prime factorisation in a cyclotomic field).

For $\mathrm{\ell }\nmid f$ the arithmetic Frobenius at every prime above $\mathrm{\ell }$ is the automorphism ${\sigma }_{\mathrm{\ell }}$ with ${\sigma }_{\mathrm{\ell }}({\zeta }_f)={\zeta }_f^{\mathrm{\ell }}$ (Arithmetic Frobenius is the power map in an unramified cyclotomic field).