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](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=&s5=a%20principal%20ideal%20theorem%20in%20the%20genus%20field&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq),
[Furuya](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=&s5=principal%20ideal%20theorems%20in%20the%20genus%20field%20for%20absolutely%20abelian%20extensions&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq))


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.