Iwasawa invariants ($\mu (M)={\sum }_i{a}_i,\lambda (M)={\sum }_j{b}_j\mathrm{deg}{f}_j$)

The invariants $\mu ,\lambda$ of a finitely generated torsion Iwasawa module measure its $p$-power elementary factors and distinguished-polynomial factors. Vanishing of $\mu$ means that the module is finitely generated over ${\mathbb{Z}}_p$, up to finite error. It does not mean that the module is finite: a factor $\Lambda /(T)$ has $\mu =0$ and is ${\mathbb{Z}}_p$.

For a finite abelian extension of $\mathbb{Q}$, the unramified Iwasawa module of its cyclotomic Zp-extension has $\mu =0$. This removes $p$-power elementary factors from its characteristic ideal. The theorem does not assert $\mu =0$ for arbitrary noncyclotomic Zp-extensions.