Euler characteristic of a one-variable Iwasawa module (${\chi }_{\Gamma }(M)=\#M_{\Gamma }/\#M^{\Gamma }=|f_M(0){|}_p^{-1}$)

Assume the invariant and coinvariant groups are finite and $f_M$ generates the characteristic ideal over ${\mathbb{Z}}_p[[S]]$, $S=\gamma -1$. The structure theorem and multiplicativity of finite Euler characteristics prove the formula; finite module errors have Euler characteristic one. If $f_M\mid f_N$ and the two Euler characteristics agree, the quotient has p-adic-unit constant term and is therefore a unit power series. Thus divisibility plus this finite arithmetic comparison upgrades to equality of characteristic ideals.