 Root-of-unity Euler system

Fix a finite excluded set $S$ containing two, and let $W_S$ contain the roots of unity of orders coprime to every prime in $S$. A root-of-unity Euler system is a nonzero algebraic-valued function $\Phi$ on $W_S$ that is even and Galois equivariant, satisfies ${\prod }_{{\rho }^q=1}\Phi (\rho \zeta )=\Phi ({\zeta }^q)$ for $q\notin S$, and satisfies $\Phi (\rho \zeta )\equiv \Phi (\zeta )(\mathrm{mod}\mathfrak{q})$ when the order of $\zeta$ is prime to $q$, for every prime $\mathfrak{q}$ over $q$. Values are required to be locally integral units where the congruence is used. These distribution and congruence properties provide auxiliary cyclotomic extensions, beyond a single norm-compatible p-power sequence.