There is a number field $L$ over $K$ where all ideals of $\mathbb{Z}_K$ are principal in $\mathbb{Z}_L$ I want to prove that there is a number field $L$ over a number field $K$ where all ideals of $\mathbb{Z}_K$ are principal in $\mathbb{Z}_L$. ($\mathbb{Z}_K$ is the ring of integers of $K$.)
This problem was given to us in an undergraduate algebraic number theory. I already found this question where localization is used but I'm trying to come up with a different solution because localization and valuations haven't been taught in the course.
Here’s my attempt: $K$ has class number $h$ so for all ideals $I = \langle \alpha, \beta \rangle$ we have $I^h = \langle \gamma \rangle $. Now let $\{\omega_1, \dots, \omega_n \}$ be an integral basis for $\mathbb{Z}_K$. So now suppose $s\alpha + t\beta \in I$. Then $(s\alpha + t\beta)^h \in \langle \gamma \rangle$ therefore $s\alpha + t\beta \in \langle  \sqrt[h]{\gamma} \rangle$. I was thinking of adding the $h$-th roots of $w_i$ to $K$ to obtain a number field. My problem is that even if we have $\sqrt[h]{\omega_i}$ for all $i$, we don't necessarily have the $h$-th root of their linear combinations. So I don't think this approach could work.
Hints would be appreciated and thanks in advance.
 A: Let $G$ be the class group of $K$ where $|G| = h$. We have equivalence classes $[I_1], \dots, [I_k]$ in $G$ that generate $G$ where each class can be represented by a true ideal $I_i \trianglelefteq \mathbb{Z}_K$. We also have $I_i^h = \langle \alpha_i \rangle$ for some $\alpha_i \in \mathbb{Z}_K$. Then we have $\sqrt[h]{I_i^h} = I_i = \sqrt[h]{\langle \alpha_i \rangle} = \langle  \sqrt[h]{\alpha_i} \rangle$. Let $L = K(\sqrt[h]{\alpha_1}, \dots, \sqrt[h]{\alpha_h})$. Clearly $L$ is a number field. We can also easily see that $\forall i \ \sqrt[h]{\alpha_i} \in \mathbb{Z}_L$. We can now check that every ideal $I \trianglelefteq \mathbb{Z}_K$ is principal in $\mathbb{Z}_L$.
We can look at the image of $I$ in $G$: $[I] = [I_1]^{e_1}[I_2]^{e_2}\dots[I_h]^{e_h}$. Then by the definition of the class group, we have $I = I_1^{e_1}\dots I_h^{e_h} \langle\theta \rangle$ for some $\theta \in \mathbb{Z}_K$. Now note that $I_i^{e_i}$ is just $\langle \sqrt[h]{\alpha_i} \rangle^{e_i}$ which is a principal ideal in $\mathbb{Z}_L$. A product of principal ideals is principal hence $I$ is principal in $\mathbb{Z}_L$.
