The field $\mathbb{Q}(\cos \frac {2\pi} {n})$ [closed]

Let $x$ be $\cos \displaystyle \frac {2\pi} {n}$ for some natural number $n$. Then is there an integer $n$ such that $\mathbb{Q}(x^2+x)\neq \mathbb{Q}(x)$? I also would like to know if there is some known algorithm or computer program to find such $n$.

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Have you taken a course in Galois theory? This would be a good homework problem for such a course, and as such is not on topic here. –  David Speyer Sep 24 '12 at 16:32
If so, I would appreciate for any hint. –  user9552 Sep 24 '12 at 17:11
One field inclusion is trivial, so I would say: search what the other means... –  Julien Puydt Sep 24 '12 at 18:02
Hint: If $\mathbb{Q}(x^2+x)$ is strictly smaller than $\mathbb{Q}(x)$, then there be some nontrivial element of $Gal(\mathbb{Q}(x)/\mathbb{Q})$ which fixes $x^2+x$. How could it happen that an automorphism fixes $x^2+x$ but not $x$? –  David Speyer Sep 24 '12 at 18:31