Cyclotomic Euler norm factor at a new prime ($N_{\mathbb{Q}({\mu }_{mq})/\mathbb{Q}({\mu }_m)}\Phi (\rho \zeta )=\Phi (\zeta {)}^{{\mathrm{Fr}}_q-1}$)

Let $q\nmid m$ be outside the excluded primes, let $\zeta$ have order $m$, and let $\rho$ be a primitive qth root of unity. Galois equivariance identifies the conjugates with $\Phi ({\rho }^i\zeta )$ for $1\le i\le q-1$. The distribution axiom includes the missing $i=0$ factor, so the field norm is $\Phi ({\zeta }^q)/\Phi (\zeta )$. Arithmetic Frobenius sends $\zeta$ to ${\zeta }^q$, giving the displayed factor. At higher powers of an already ramified prime all q translates occur instead. These are different norm relations; norm compatibility in just a p-power tower does not supply the auxiliary-prime relation by itself.