<s>[INST] Factor polynomial over $C, Q, R$, if one comlex root is given The polynomial is: $P(x)=x^6+x^4-x^3+x^2+1$. I need to factor it over $C, Q, R$ if one complex root is $\sqrt[3]{1}$. Also find all fields in which $P$ is reducible. 
Now, I know how to find one factor of P using given complex root, and I end up with $(x^2+x+1)(x^4-x^3+x^2-x+1)$, but how should I go about $(x^4-x^3+x^2-x+1)$? 
How can I find all fields in which $P$ is reducible?
 [/INST] A start: To factor the quartic, it may be useful to find its roots.
Rewrite the quartic as $x^2(x^2+\frac{1}{x^2}-\left(x+\frac{1}{x}\right)+1)$.
Let $t=x+\frac{1}{x}$. Then to solve our quartic we first solve
$$t^2-2-t+1=0.$$
Another way: Our quartic is $\frac{x^5+1}{x+1}$. So its roots are certain $10$-th roots of unity.
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