= Cyclotomic reduction at a conductor prime
{title2=$\Phi_{p^a m}\equiv\Phi_m^{\varphi(p^a)}\pmod p,\quad p\nmid m$}
For $a\ge1$ 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> $\varphi(p^a)$ 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.
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