Iwasawa class-number growth theorem ($v_p(h_{F_n})=\mu p^n+\lambda n+\nu (n\gg 0)$)

For the finite layers of a Zp-extension of a number field, the p-primary ideal class groups eventually satisfy the displayed formula with integer invariants $\mu ,\lambda \ge 0$ and $\nu \in \mathbb{Z}$. The first two are the Iwasawa invariants of the unramified Iwasawa module. The Iwasawa module structure theorem and control of the finite-layer norm and ramification terms give the result. For a cyclotomic extension of an abelian field, the Ferrero-Washington theorem gives $\mu =0$. This growth theorem records two invariants; it is weaker than identifying the full characteristic ideal in the Iwasawa main conjecture.