= Iwasawa class-number growth theorem
{c}
{title2=$v_p(h_{F_n})=\mu p^n+\lambda n+\nu\quad(n\gg0)$}
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 $\mu,\lambda\geq0$ 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>.
Back to article page
