TITLE: Splitting of primes in cyclotomic $\mathbb{Z}_p$-extension
QUESTION [6 upvotes]: Let $p$ be a prime, $K$ be a finite extension of $\mathbb{Q}$ and $K_{\infty}$ be a cyclotomic $\mathbb{Z}_p$-extension of $K$ i.e. Gal$(K_{\infty}/K) \cong \mathbb{Z}_p$, the group of $p$-adic integers under addition. 
Then

What can we say about the prime decomposition in $K_{\infty}$ for any prime of $\mathbb{Q}$ for e.g. ramification index, order of the decomposition subgroup, finitely decomposed or not ?

The only result I know in this direction is that such extensions are unramified outside of $p$. 

Can we say something more when $K=\mathbb{Q}$ ?

REPLY [5 votes]: For any $\mathbb{Z}_p$-extension the ramification is concentrated among the primes above $p$. Those that are ramified are totally ramified.
For the cyclotomic $\mathbb{Z}_p$-extension all places above $p$ are totally ramified. That is because you obtain it as a subextension of $\bigcup_n K(\mu_{p^n})$. For all unramified places the decomposition group is of finite index.