Cube root of unity again

Algebra Level 3

Let $w$ be cube root of unity other than 1. Find the sum of all pairs of integers $(a,b)$ such that

$(aw + b)(aw^2 + b) = 1$

-3 0 3 1

1 solution

Dev Sharma
Oct 16, 2015

The equation is same as :

$a^2 + b^2 - ab - 1 = 0$

$D = 4 - 3b^2 > 0$

So pairs are (1,0),(-1,0),(0,-1)(-1,-1)(0,1)(1,1)

And their sum is 0.