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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 $K=\mathbb{Q}$ ?

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    $\begingroup$ 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. $\endgroup$ Jul 31 '18 at 17:14
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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.

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  • $\begingroup$ Kindly provide some references. $\endgroup$
    – Robert
    Jul 31 '18 at 10:24
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    $\begingroup$ The first part is in Washington "Introduction to cyclotomic fields" chapter 13.The second follows from Galois theory and ramification and prime decomposition in cyclotomic extension as in Theorem 2.13 in Washington. $\endgroup$ Jul 31 '18 at 10:56

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