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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.

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

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

Let $E$ be the maximal unramified 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 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 an 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.

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 $\operatorname{Cl}_K$, namely 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 becomes principal in $E$."

I tried to modify the method of ibid., 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.

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.

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, 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, 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.

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 $\operatorname{Cl}_K$, namely 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 becomes principal in $E$."

I tried to modify the method of ibid., 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.