= Euler characteristic of a one-variable Iwasawa module
{c}
{title2=$\chi_\Gamma(M)=\#M_\Gamma/\#M^\Gamma=|f_M(0)|_p^{-1}$}
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.
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