 Solution

For $x\in {\mathcal{O}}_K$, all $\mathrm{Tr}({\alpha }^{i-1}x)$ are integers, and duality gives $x={\sum }_i\mathrm{Tr}({\alpha }^{i-1}x)y_i$. Hence ${\mathcal{O}}_K\subset \sum \mathbb{Z}{y}_i$. 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.