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



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



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> Can we say something more when $K=\mathbb{Q}$ ?
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Splitting of primes in cyclotomic $\mathbb{Z}_p$-extension
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.


