Question 133
a) Let $z$ be a complex number. Prove that if $w$ is a cube root of $z$, then the numbers $\varepsilon \cdot w$ and $\varepsilon^{2} \cdot w$ are the other cube roots of $z$, where $\varepsilon=-\frac{1}{2}+i \frac{\sqrt{3}}{2}$.
b) Let $a, b, c \in \mathbf{C}$ be pairwise distinct such that $(a+b)^{3}=(b+c)^{3}=(c+a)^{3}$. Prove that $a^{3}=b^{3}=c^{3}$.
b) Let $a, b, c \in \mathbf{C}$ be pairwise distinct such that $(a+b)^{3}=(b+c)^{3}=(c+a)^{3}$. Prove that $a^{3}=b^{3}=c^{3}$.