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Show that there are no $a,b,c,d\in \mathbb{Q}(i)$ such that $e^{2\pi i/5}=a+b\sqrt[4]{2}+c\sqrt{2}+d(\sqrt[4]{2})^{-1}$
Previously in this problem, I found all the subfields of $\mathbb{Q}(\sqrt[4]{2},i)$ with the help of the subgroups of $D_4$ and Galois's Theorem as you can see in the followin...