# 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

 $N_{\mathbb{Q}(\omega)}^{\mathbb{Q}}(a+b\omega)=(a+b\omega)(a+b\omega^{2})=a^{2% }-ab+b^{2},Tr_{\mathbb{Q}(\omega)}^{\mathbb{Q}}(a+b\omega)=(a+b\omega)+(a+b% \omega^{2})=2a-b$
Title examples of trace and norm ExamplesOfTraceAndNorm 2013-03-22 15:55:45 2013-03-22 15:55:45 polarbear (3475) polarbear (3475) 16 polarbear (3475) Example msc 12F05