<s>[INST] How to see $x^6-1=(x^2−1)(x^2+x+1)(x^2−x+1)$? I was reading an example where the purpose was to compute a certain Galois group. Along the way, the writer says : note $x^6-1=(x^2−1)(x^2+x+1)(x^2−x+1)$. But how do I note this? I understand you can factorize by $x^2-1$, since when I draw on the unit circle I see that $-1$ and $+1$ are roots. But for the rest?
Edit :
I see you can then factor $(x^2-1)(x^4+x^2+1)$ and than substitute $x^2=y$ and solve quadratic equation but can you actually see the solution visually?
 [/INST] Also, by using your idea we obtain:
$$x^6-1=(x^2-1)(x^4+x^2+1)=(x^2-1)(x^4+2x^2+1-x^2)=$$
$$=(x^2-1)((x^2+1)^2-x^2)=(x^2-1)(x^2-x+1)(x^2+x+1).$$
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