= Root-of-unity Euler system
= Three-axiom Euler system
{synonym}
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 $\Phi(\rho\zeta)\equiv\Phi(\zeta)\pmod{\mathfrak q}$ 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.
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