Exercise 424Algebra › Complex numbersHigh schoolAdded 3 Nov 2026

The cube roots of unity

#cube roots#unity#cyclic

Statement

Find the cube roots of unity in algebraic form.

Hint

Solve $z^3=1$ , or note $z^3-1=(z-1)(z^2+z+1)$ .

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

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

Quadratic roots. $z=(-1\pm i\sqrt3)/2$ .

Notation. Let $j=e^{2i\pi/3}=(-1+i\sqrt3)/2$ . Then $\bar j=j^2=(-1-i\sqrt3)/2$ .

Conclusion. The cube roots of unity are $1, j, j^2$ , with $1+j+j^2=0$ .