I'm looking for $\alpha$ such that $\mathbb{Q}\subset \mathbb{Q}(\sqrt{17})\subset \mathbb{Q}(\alpha)\subset \mathbb{Q}(\zeta_{17}+\zeta_{17}^{-1})\subset \mathbb{Q}(\zeta_{17})$ will be a chain of extensions with extension degree of 2 for every extension, when $\zeta_{17}=e^{2\pi i/17}$. How do I approach that? (need a direction, not solution. thanks)
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Do you know how to construct the minimum polynomials of $\sqrt{17}$ and $\zeta_{17}+\zeta_{17}^{-1}$? – Algebear Apr 30 '18 at 19:17
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You can approach this step by step. The last extension has degree $2$ for an easy reason, see here. The first one, too, because $x^2-17$ is irreducible. So there is not too much left. – Dietrich Burde Apr 30 '18 at 19:22
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Gaussian periods? – Angina Seng Apr 30 '18 at 19:35
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yes, I just built the polynoms. seems to work. and I looked for Gaussian periods in Wikipedia. Looks very similar to my problem, but I'm not sure what did you want to say about them. @LordSharktheUnknown can you clarify? thanks – S. R Apr 30 '18 at 19:59