Let $\displaystyle K=\mathbb{Q}(\sqrt[n]{a})$, where $\displaystyle a \in \mathbb{Q}, a>0$ and suppose $\displaystyle [K:\mathbb{Q}]=n$.

Let $\displaystyle E$ be any subfield of $\displaystyle K$ and let $\displaystyle [E:\mathbb{Q}]=d$.

Prove that $\displaystyle E=\mathbb{Q}(\sqrt[d]{a})$.