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

$${N}_{\mathbb{Q}(\omega )}^{\mathbb{Q}}(a+b\omega )=(a+b\omega )(a+b{\omega}^{2})={a}^{2}ab+{b}^{2},T{r}_{\mathbb{Q}(\omega )}^{\mathbb{Q}}(a+b\omega )=(a+b\omega )+(a+b{\omega}^{2})=2ab$$ 
