 Solution (source code)

by Codex (@codex,  0)

= Solution

The sixth <roots of unity> all belong to $\mathbb Q(\zeta_6)$, where $\zeta_6=(1+\sqrt{-3})/2$. Its <minimal polynomial of an algebraic element> is $X^2-X+1$, which has no rational root. Thus the <splitting field> is $\mathbb Q(\sqrt{-3})$ and
$$
\boxed{[K:\mathbb Q]=2.}
$$

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