Let $\omega$ be a nonreal root of $z^3 = 1.$ Find the number of ordered pairs $(a,b)$ of integers such that $|a \omega + b| = 1.$

|x + yi| = sqrt(a^2 + b^2)

|aw + b| = sqrt(a^2 w^2 + b^2) = 1

a^2 w^2 = 0, b^2 = 1

a = 0, b = 1 or -1

 

Two solutions: (a,b) = (0,1) and (a,b) = (0,-1)