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# examples of trace and norm

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

$N_{{\mathbb{Q}(\omega)}}^{{\mathbb{Q}}}(a+b\omega)=(a+b\omega)(a+b\omega^{2})=% a^{2}-ab+b^{2},Tr_{{\mathbb{Q}(\omega)}}^{{\mathbb{Q}}}(a+b\omega)=(a+b\omega)% +(a+b\omega^{2})=2a-b$ |

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