Let $\omega$ be a complex root of unity different than 1. Then $\omega$ and $\omega^2$ are the conjugate roots of the minimal polynomial $x^2 + x +1$ Since $\mathbb{Q}(\omega)$ is the splitting field of $x^2 + x +1$ it is Galois over $\mathbb{Q}$ Moreover the Galois group $Gal(\mathbb{Q}(\omega)/\mathbb{Q}))$ is formed by the identity and the automorphism $g(\omega) = \omega^2$ The elements of $\mathbb{Q}(\omega)$ have the form $a + b\omega$ $a,b\in \mathbb{Q}$ Then we obtain $$N_{\mathbb{Q}(\omega)}^{\mathbb{Q}}(a + b\omega) = (a +b\omega)(a +b\omega^2) = a^2 - ab + b^2, Tr_{\mathbb{Q}(\omega)}^{\mathbb{Q}}(a + b\omega) = (a +b\omega) + (a + b\omega^2) = 2a - b $$