# nt.number theory – From which layer can we apply the class number formula of Iwasawa in the extension Z_p?

To let $$K$$ to be a number field. To let $$K_ infty / K$$ be a $$mathbb {Z} _p$$ Extension. Iwasawa proved that there are four integers $$n_0, mu, lambda, nu geq 0$$ so for everyone $$n geq n_0$$, $$mathrm {ord} _p (h_n) = mu p ^ n + lambda n + nu,$$

from where $$h_n$$ is the class number of $$n$$layer field $$K_n$$ (so $$K_0 = K$$).

My question is what we know about $$n_0$$? If we know how to calculate $$n_0$$then we can count $$mu, lambda, nu$$ in finite steps, at least theoretically.

It's hard for me to find an explicit formula for $$n_0$$ due to the pseudo-isomorphism.

One may ask that this is the whole number $$r_0$$ so that $$K_ infty / K_ {r_0}$$ is completely branched out with each branched main work? That's wrong. There are many examples in the work of Schoof-Kraft [Computing Iwasawa modules for real quadratic number fields], $$K_ infty / K_0$$ is completely branched at every branched prime, but $$n_0> 0$$,