Factorisation of polynomials

Question

A friend of mine asked me this question on factorisation. We have the expression $$x^2-1$$ $$=(x+1)(x-1)$$ Now if we consider the identity $x^3+y^3=(x+y)(x^2-xy+y^2)$, can we write $x+1$ as $(x^{1/3})^3 +(y^{1/3})^3$. If yes then can we consider its factors to be also a factor of $x^2-1$. Or can these identities only be used in case of real numbers? Does this have any connections with unique factorisation domains?

Answer 1

Of course you can use the identities in general, i.e. not only for real numbers. If I understand you correctly, then what you propose is $$ x + y = (x^{1/3})^3 +(y^{1/3})^3=(x^{1/3}+y^{1/3})(x^{2/3}-(xy)^{1/3}+y^{2/3}) $$ Then you further want to use this with $y=1$ and $y=-1$ to impose $$ x^2-1 = (x+1)(x-1)=\\ =(x^{1/3}+1)(x^{2/3}-x^{1/3}+1)(x^{1/3}+(-1)^{1/3})(x^{2/3}-(-x)^{1/3}+(-1)^{2/3})=\\ =(x^{1/3}+1)(x^{2/3}-x^{1/3}+1)(x^{1/3}+e^{i \pi/3})(x^{2/3}-e^{i \pi/3}x^{1/3}+e^{i 2 \pi/3}) $$

All of that is fine, but you cannot use it for any further insights on factorisation.