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Let $x$ be $\cos \displaystyle \frac \pi n$ for some natural number $n$. Then is it true that $\mathbb{Q}(x^2+x)=\mathbb{Q}(x)$?

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    Well, of course the inclusion in one direction is easy. What are your thoughts on the other direction: $x \in \mathbb Q(x^2+x)$ ??2012-09-17
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    I don't have any good idea. We can consider the field $\mathbb{Q}(cos \frac \pi n)$ as the maximal real subfield of the cyclotomic field $\mathbb{Q}(\zeta_{2n})$.2012-09-17

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