I'm struggling to figure out how to find the roots of this question.
If $w$ is a complex root of the equation: $$x^7-1=0$$ show that $w^2,w^3,w^4,w^5, w^6$ are the other complex roots.
I know that you can factorise it into:
$$(x-1)(x^6+x^5+x^4+x^3+x^2+x+1)=0$$
But I don't know where to go from there.
Any help is greatly appreciated!
Best Answer
Let $w$ be a complex root of $x^7-1=0$. Then $w^7=1$ and $w\ne1$.
Now $w^n$ is also a root of $x^7-1=0$ because $(w^n)^7=w^{7n}=(w^7)^n=1^n=1.$
Furthermore, if $n$ is relatively prime to $7$ then $w^n\ne1$, or else we would have $w=1$,
and that would contradict the assumption that $w$ is complex.