I usually consider a cyclic extension $K$ of degree an odd prime $p$ over the rational field $\mathbf{Q}$.

In this case, there is well-known result that "every ambiguous class in the class group $\operatorname{Cl}_K$, which is a class which is fixed by the Galois group of $\operatorname{Gal}(K/\mathbf{Q})$, becomes trivial in the genus field of $K/\mathbf{Q}$" (ref. Terada, Furuya)

Let $E$ be the maximal unramifield extension of $K$ whose Galois group over $K$ is isomorphic to an elementary abelian $p$-group. In my opinion, the analogue of the principal ideal theorem should hold for $E/K$. More precisely, I guess the following statement is true:

"Every ideal $I$ of $K$ such that $I^p$ is a principal (in $K$) becomes principal in $E$."

I tried to modify the method of Terada or Furuya, and failed. (Because, in some sense, a symmetric of the form of $I^p$ is less than $I^ {(1-\sigma)}$ where $\sigma$ is a generator of $\operatorname{Gal}(K/\mathbf{Q})$ and there is explicit expression about the genus field but not $E$.)

If you know about a method for this type of problems, related good references, counter-examples, or have some comments, please let me know.

Sorry for my poor English. Thank you for your attention.