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

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

Title	examples of trace and norm
Canonical name	ExamplesOfTraceAndNorm
Date of creation	2013-03-22 15:55:45
Last modified on	2013-03-22 15:55:45
Owner	polarbear (3475)
Last modified by	polarbear (3475)
Numerical id	16
Author	polarbear (3475)
Entry type	Example
Classification	msc 12F05