In iwasawa theory, we often suppose $\Bbb{Z}_p-$ extension satisfies the following condition.
・Exactly one prime is ramified
・And the prime is totally ramified
Which $\Bbb{Z}_p-$ extension satisfies this two condition ? In particular ,I want to know the reason why $\Bbb{Z}_p-$ extension over $\Bbb{Q}(ζ_p)$ satisfies the condition. Thank you in advance.
Back ground: I want to check iwasawa class formula holds for some $\Bbb{Z}_p-$ extension, and Washington's book 'Introduction to cyclotomic fields',$Prop13.22$ supposes above two assumption, if I could check this assumption, I will be able to check iwasawa class number formula for that field.
It is not hard to see that $\Bbb{Q}(\zeta_{p^\infty})/\Bbb{Q}$ is ramified only at $p$ where it is totally ramified, and that for $p$ odd its Galois group is $\Bbb{Z}_p^\times \cong C_{p-1}\times \Bbb{Z}_p$ (for $p=2$ it is $\Bbb{Z}_2^\times \cong C_2\times \Bbb{Z}_2$), so $$\Bbb{Q}(\zeta_{p^\infty})/\Bbb{Q}(\zeta_p) \text{ and } \Bbb{Q}(\zeta_{p^\infty})^H/\Bbb{Q}$$ both satisfy the requirements, where $H= \{ h_a: \zeta_{p^n}\mapsto \zeta_{p^n}^{a^{p^n}}, a\in 1\ldots p-1\}$ for $p$ odd (for $p=2$ it is $H= \{ h_a: \zeta_{p^n}\mapsto \zeta_{p^n}^a, a=\pm 1\}$)