In their classic paper "Class fields of abelian extensions of $\mathbf{Q}$", Mazur and Wiles assert that

"in a cyclotomic $\mathbf{Z}_p$-extension only finitely many primes lie above any prime of $\mathbf{Q}$."

My only other source in learning this material so far has been Washington's "Introduction to Cyclotomic Fields", and the only result along these lines is that such extensions are unramified outside of $p$.

So apparently, all primes lying above $l \neq p$ stop splitting at some finite level $K_n$, after which they remain inert. I've been unable to make much progress is proving this.

How can we see that this statement is true, and what other, more general results do we have about prime decomposition in cyclotomic extensions?


1 Answer 1


The point is that the Frobenius at $\ell$ is nontrivial in this extension, so it generates an open subgroup, and the fixed field of this subgroup is precisely the field at which $\ell$ stops splitting.

To see that Frob$_\ell$ is nontrivial, recall that in the full $\mathbb{Z}_p^*$ extension $\mathbb{Q}(\mu_{p^\infty})$, Frob$_\ell$ is simply the element $\ell\in \mathbb{Z}_p^*$, which is clearly nontrivial. The cyclotomic $\mathbb{Z}_p$ extension is obtained by quotienting $\mathbb{Z}_p^*$ by the subgroup $\mu_{p-1}$. This subgroup does not contain $\ell$, so Frob$_\ell$ remains nontrivial in the $\mathbb{Z}_p$-extension.

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    $\begingroup$ By the way, this need not hold for an arbitrary $\mathbb{Z}_p$-extension of a number field. For instance, suppose $K$ is a number field which has a $\mathbb{Z}_p^2$-extension $K_\infty$. Any prime $\ell$ will split completely in the fixed field of Frob$_\ell$, and the galois group of the fixed field over $K$ has $\mathbb{Z}_p$-rank at least one. $\endgroup$ Sep 15, 2011 at 19:59
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    $\begingroup$ A minor thing: the subgroup Frobenius <<topologically generates>> is open (being closed and non-trivial in $\mathbf{Z}_p$, hence closed and finite index). $\endgroup$ Sep 16, 2011 at 23:25

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