Let $p$ be a prime and let $C = \mathbb{F}_p^\times$. Then Gal$(\mathbb{Q}(\zeta_p)/\mathbb{Q})$, where $\zeta_p$ is a primitive $p$th root of unity, may be identified with $C$ in the obvious way. Let $\theta_p$ and $J_p$ denote the Stickelberger element and the Stickelberger ideal in $\mathbb{Z}C$, respectively. The integral group ring $\mathbb{Z}C$ acts on the class group Cl$(\mathbb{Q}(\zeta_p))$ in the obvious way, and the classical Stickelberger's theorem says that $J_p$ annihilates Cl$(\mathbb{Q}(\zeta_p))$.

Now say $K\neq \mathbb{Q}$ is an arbitrary number field such that $[K(\zeta_p):K]=p-1$. Then the discussion above still makes sense, but in general Cl$(K(\zeta_p))$ is not annihilated by $J_p$.

Question: For a fixed $K\neq\mathbb{Q}$, is there always at least one prime $p$ such at Cl$(K(\zeta_p))$ is not annihilated by $J_p$?

I think this is a difficult question. But any thoughts or references related to this would be appreciated. As a side note, this question came up as I was thinking about a problem on Galois module structure of rings of integers; they are related by the work of L.R. McCulloh.

ADDED: For a fixed prime $p$, I think it is sufficient to show that there is a prime $\ell$ that divides the order of the minus class group of $K(\zeta_p)$ but not that of $\mathbb{Q}(\zeta_p)$.

Does anyone know anything about this?


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