Solving $z^3=1$

#cube roots of unity#algebraic solution

Statement

Solve in $\mathbb{C}$ the equation $z^3=1$ .

Hint

Factor $z^3-1=(z-1)(z^2+z+1)$ .

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Factor. $z^3-1=(z-1)(z^2+z+1)$ .

Solutions. $z=1$ or $z^2+z+1=0$ .

Quadratic. $z=\frac{-1\pm i\sqrt3}{2}$ .

Conclusion. $z\in\{1,\frac{-1+i\sqrt3}{2},\frac{-1-i\sqrt3}{2}\}$ .