To find the degree of extension $[\Bbb Q(\zeta_9):\Bbb Q(\zeta_9+\zeta^{-1}_9)]$, I need to formally conclude that $ x^2-(\zeta_9+\zeta^{-1}_9)x+1 $ is irreducible over $\Bbb Q(\zeta_9+\zeta^{-1}_9)$. And then I got stucked. Could someone please help? Thanks in advance!
2026-04-08 12:34:55.1775651695
How to formally conclude that $ x^2-(\zeta_9+\zeta^{-1}_9)x+1 $ is irreducible over $\Bbb Q(\zeta_9+\zeta^{-1}_9)$
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$\zeta$ is not real. (Nor is $\zeta^{-1}$.)