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I'm struggling with a question I've got as I can't seem to wrap my head around Galois theory much.

i)Prove that $\zeta^{2}$ is a primitive 8th root of unity and $\zeta^{4}$ is a primitive 4th root of unity. Deduce then that $\zeta$ is a constructible number, where $\zeta$ is primitive 16th root of unity.

I've tried different methods for this now and I'm just confusing myself more unfortunately. We had a question earlier which I was trying to work off since they seemed similar but now I'm not sure. The question was

  • If $\zeta$ is the primitive 10th root of unity then show $-\zeta$ is a 5th root of unity.

They answered this by saying:

$1=\zeta^{16}=(\zeta^{5})^2$, thus $\zeta^{5}$ is one of $\pm$1. Since $\zeta$ is a primitive 10th root of unity then $\zeta^{5}=1$. Thus: $(-\zeta)^5=(-1)^5\zeta^{5}=(-1)(-1)=1$ (eqn.2), which shows its a 5th root and the fact that $-\zeta \neq 1$ we get that $-\zeta$ is primitive 5th root of unty.

I have been trying to apply this to my question as its the only similar example weve been given but i cant get my eqn.2 to equal 1.

Am I going about this in the worst way/ just a wrong way? Any help would be appreciated since Galois theory is very tricky for me.

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    $\begingroup$ It may be worth noticing that the last part off the question, about constructibility, could be done without Galois theory, if it didn't insist on "Deduce". Just using elementary geometry, specifically the fact that you can bisect angles with ruler and compass. Bisect one of the four quadrants of the plane; then bisect one of the two resulting 45-degree angles; then intersect that ray with the unit circle. $\endgroup$ Nov 25, 2021 at 2:07

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You can prove this directly from definitions. Obviously $\zeta^2$ is an $8$th root of unity and $\zeta^4$ is a $4$th root of unity, so all you need to show is that they're primitive. Suppose they weren't. That would mean, in the one case, that $\zeta^2$ is a $4$th root of unity, which would mean that $\zeta$ is an $8$th root of unity, which would mean that $\zeta$ is not a primitive $16$th root of unity, contrdicting our choice of $\zeta$. An analogous argument works for $\zeta^4$.

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