Let ω be a primitive 7th root of unity in $\Bbb C$ and set $α := ω + ω
^6$
. Determine, with
justification, the minimum polynomial of α over Q.
Would one use logs in such a question , or how should one begin? it's the roots of unity that are throwing me off, I know how to these minimal polynomial questions for radicals etc..
Best Answer
One has the following $\omega^7=1$ and $\omega^6+\omega^5+\omega^4+\omega^3+\omega^2+\omega+1=0$ Now we compute
$$\alpha=\omega+\omega^6$$ $$\alpha^2=\omega^2+\omega^5+2$$ $$\alpha^3=\omega^3+\omega^4+3\alpha$$
Adding the three identities and rearranging one gets
$$\alpha^3+\alpha^2-2\alpha-1=0$$
And $X^3+X^2-2X-1$ is irreducible over $\Bbb{Q}$ because it has no integer roots and therefore no rational roots (it is monic). So we have the minimal polynomial of $\alpha$ over $\Bbb{Q}$