Cyclotomic reduction at a conductor prime (${\Phi }_{p^{a}m}\equiv {\Phi }_m^{\phi (p^a)}(\mathrm{mod}p),p\nmid m$)

For $a\ge 1$ and $p\nmid m$, the identity ${\Phi }_{p^{a}m}(X)={\Phi }_m(X^{p^a})/{\Phi }_m(X^{p^{a-1}})$ gives the displayed reduction, since taking a $p$th power equals substitution $X\mapsto X^p$ in characteristic $p$. The reduction of ${\Phi }_m$ is squarefree because it divides the separable polynomial $X^m-1$. The Kummer-Dedekind theorem therefore gives ramification index of a prime ideal $\phi (p^a)$ in the cyclotomic field. In particular, a conductor factor $2$ occurring only to the first power has index one and creates no ramification by itself.