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Given $F\subseteq \mathbb{C}$ be the splitting field of $x^7-2$ over $\mathbb{Q}$ and $z=e^{2\pi i/7}$, a primitive seventh root of unity. Since it is seventh root of unity what is the degree of the extension $[F:\mathbb{Q}(z)]=?$. I tried with the Cyclotomic extension kind of result but there it is not explicitly mentioned what will be the degree when the extension is over splitting field. Help me out.

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Obviously $F=\mathbb{Q}(\sqrt[7]{2},z)$. Now $[\mathbb{Q}(\sqrt[7]{2}):\mathbb{Q}]=7$ and $[\mathbb{Q}(z):\mathbb{Q}]=6$, so $[F:\mathbb{Q}]=42$ (it is at most $42$ because $[F:\mathbb{Q}(\sqrt[7]{2})]$ is at most $6$, and it must be divisible by both $6$ and $7$).

From here we also get $[F:\mathbb{Q}(z)]=7$.

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    $\begingroup$ Very clear answer, but I know that when I was studying this, diagrams helped me immensely. Therefore: this $\endgroup$ – Krijn Sep 20 '15 at 12:44

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