I usually consider a cyclic extension ***K*** of degree an odd prime p over the rational field ***Q***.

In this case, there is well-known result that
"every ambiguous class ideal whose class is fixed by the Galois group of
***K*** over ***Q*** can be a principal ideal in the genus field of ***K***/***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, for $E$, there is an analogue of principal ideal theorem. More precisely, I guess the following statement is true:
"Every ideal ***I*** of ***K*** such that ***I*** ^ p is a principal can be principal ideal in ***E***."

I tried to modify the method of above ref., and failed.
(Because, in some sense, a symmetric of the form of ***I*** ^ p is less than ***I*** ^ (1-*s*) where *s* is a generator of ***Gal(K/Q)*** and  there is explicit expression about the genus field but not ***E****.)


If you know about a method of this type problem,  related good reference,  counbter exapmle, or have some comments, please let me know.

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