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My script states that there are $2$ non-trivial subfields of $\mathbb{Q}(\zeta_7)$, where $\zeta_7$ is a $7$-th primitive root of unity. These subfields are $\mathbb{Q}(\zeta_7 +\overline \zeta_7)$ and $\mathbb{Q}(\sqrt{-7})$.

Definition of radical extension I am using:

An extension $L/K$ is a radical extension, if $\exists\gamma_1,...,\gamma_r\in L, n_1,...,n_r\in\mathbb{N}$, such that $L=K(\gamma_1,...,\gamma_r)$ and $\gamma_i^{n_i}\in K(\gamma_1,...,\gamma_{i-1})\ \forall i$

Ok, so using the definition on $\mathbb{Q}(\zeta_7)$ I obviously have the first condition satisfied and the second with $\zeta_7^7=1\in\mathbb{Q}$.

I am not sure what I don't understand at this point. Also I am unsure of why the subfields are even mentioned. In my script the lattice of the subfields is drawn at it says (translated from german): "Not a radical extension because $\zeta_7$ is only present at the the top of the lattice". I have no idea what this means.

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  • $\begingroup$ @reuns I am sorry, I cannot follow your comment. Do you mean "...hence so is $\mathbb{Q}(\zeta_7,\zeta_3)/\mathbb{Q}(\zeta_3)$? I am unfamiliar with Kummer-Theory, I am hearing a basic algebra class. Why do I need $\zeta_3$? I just want a simple radical extension $\mathbb{Q}(\zeta_7)/\mathbb{Q}$. $\endgroup$ – EpsilonDelta Feb 25 at 12:37
  • $\begingroup$ $\mathbb{Q}(\zeta_7)/\mathbb{Q}$ is a $7$-radical extension needing a $7$-root of unity, $\mathbb{Q}(\zeta_7,\zeta_3)/\mathbb{Q}$ is a $2,3$-radical extension not needing any root of unity, and none of them is constructible (ie. a $2$-radical extension). $\endgroup$ – reuns Feb 25 at 12:39
  • $\begingroup$ Every cyclotomic field is both Galois and radical over $\Bbb Q$, see here. $\endgroup$ – Dietrich Burde Feb 25 at 12:48

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