Solution
= Solution
For $x\in\mathcal O_K$, all $\operatorname{Tr}(\alpha^{i-1}x)$ are integers, and duality gives $x=\sum_i\operatorname{Tr}(\alpha^{i-1}x)y_i$. Hence $\mathcal O_K\subset\sum\mathbb Zy_i$. Every <ideal> of $R$ is therefore a <subgroup> of a finitely generated free abelian <group> and is finitely generated over $\mathbb Z$, hence over $R$; so $R$ is Noetherian.