The fourth roots of 1 are x^4=1 or x^4-1=0. how many fourth roots of 1 are there?

Factoring difference of squares:       we see that we have real roots of  , .but  can be factored over the complex numbers. since , we have    so we also have complex roots of  ,  .we have a total of 4 fourth roots of 1 (2 real roots, 2 complex roots).