Let $a_n:=e^{2\pi i/n} $ , then how to show that $[\mathbb Q(a_n):\mathbb Q(a_n +1/a_n)]=2$ ? From this and using Galois theory can we determine which $a_n$ are irrational ?
2026-03-25 22:30:53.1774477853
$a_n:=e^{2\pi i/n} $ , then how to show that $[\mathbb Q(a_n):\mathbb Q(a_n +1/a_n)]=2$ ?
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Hint:
$p(x) = x^2 - (a_n + \frac 1 {a_n})x + 1$