Cube Root of Unity

(i) The cube roots of unity are 1, (-1+i√3)/2, (-1-i√3)/2 .

(ii) If w is one of the imaginary cube roots of unity then 1+w+w²=0. In general 1 + wⁿ+w²ⁿ=0; where n ε  I but is not the multiple of 3.

(iii) In polar form the cube roots of unity are:

        cos 0 + i sin 0; cos2Π/3 + isin2Π/3; cos4Π/3 + isin4Π/3.

(iv) The three cube roots of unity when plotted on the argand plane constitutes the vertices of an equilateral triangle.

(v) The following factorization should be remembered:

        (a, b, c ε R and w is the cube root of unity)

        a³ - b³= (a-b) (a-wb) (a-x²b);

        x² +x+1=(x-w)(x-w²);

        a³ + b³= (a+b) (a+wb) (a+x²b);

        a² +ab+b²=(a-bw)(a-bw²)

        a³ + b³ + c³ - 3abc = (a+b+c) (a+wb+w²c) (a+w²b+wc);

n^th Roots of unity -

if 1, α₁,  α₂,......α_n-1 are the n, nth roots of unity then: