To me it is seemingly true, e.g. $\mathbb{Q}(\sqrt2))$,$\mathbb{Q}(\sqrt[3]2))$ are all examples. Also, $$\mathbb{Q}(\sqrt{2}+\sqrt{3})= \mathbb{Q}(\sqrt{2},\sqrt{3})=\mathbb{Q}(\sqrt{2})(\sqrt{3})$$ Then we can see that it is a radical extension $$ \mathbb{Q}\subseteq \mathbb{Q}(\sqrt{2})\subseteq(\mathbb{Q}(\sqrt{2}))(\sqrt{3}) $$ Even though the numbers are coupled, we can decouple them.

Actually, it is not very easy (for me) to prove some extension is not radical. Because we have to run over all possible towers of fields. Is there any examples?


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