Factoring completely using complex cube of unity

Question

How can you completely factor $a^2 + ab + b^2$ and $a^2 - ab + b^2$ completely using $\omega$, the complex root of unity? Is there some general rule for such complex factorisations? Any help would be appreciated.

Answer 1

Homogeneous polynomials -- polynomials where every monomial has the same total degree -- behave in many respects like ordinary polynomials in one fewer variable. So your two polynomials act like polynomials in one variable, so to factor them you just need to find the roots.

Some various ways to think of this is

• think of them as quadratic polynomials in $a$ with $b$ just being some constant
• factor out $b^2$, leaving you with a quadratic in $(a/b)$
• make a change of variable $a = b \cdot c$.