Solution
= Solution
All sixth roots of unity lie in $\mathbb Q(\zeta_6)$, with $\zeta_6=(1+\sqrt{-3})/2$. Its quadratic polynomial $x^2-x+1$ has no rational root, so
$$
\boxed{K=\mathbb Q(\sqrt{-3}),\qquad[K:\mathbb Q]=2}.
$$
= Solution
All sixth roots of unity lie in $\mathbb Q(\zeta_6)$, with $\zeta_6=(1+\sqrt{-3})/2$. Its quadratic polynomial $x^2-x+1$ has no rational root, so
$$
\boxed{K=\mathbb Q(\sqrt{-3}),\qquad[K:\mathbb Q]=2}.
$$