# Splitting of primes in cyclotomic $\mathbb{Z}_p$-extension

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 $K=\mathbb{Q}$ ?

• You say "a" cyclotomic extension, but beware that there exists a unique cyclotomic extension for every number field $K$ ($p$ being fixed, of course). For this one, the decomposition of primes can be deduced, as Chris says in his answer, from the decomposition of primes in $\mathbb{Q}(\zeta_{p^\infty})/\mathbb{Q}$: and this shows in passing that every prime is finitely split, and the answer to your final question is "yes, since $\mathbb{Q}$ admits a unique $\mathbb{Z}_p$-extension, precisely the cyclotomic one". For general $\mathbb{Z}_p$-extensions, there might exist infinitely split primes. Jul 31 '18 at 17:14

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.