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.


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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