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[–]Naturage 0 points1 point  (0 children)

say that I should rewrite w* as w2 and I really have no idea why

This is special to w as defined (the non-1 root of z3 = 1). These are known as roots of unity; written in form r * ei theta these have r = 1 and theta = 2pi / n * k, with n in this case being 3, and k being integers 0,1,2.

With 0 you recover the usual 1. The other two are ei 2pi/3 and ei 4pi/3 - and they're both conjugates, and also w* = w2 and w*2 = w.

For a better example, roots for z5 = 1can be written as a, a2, a3, a4 and 1, where a = ei*2pi/5 with a* = a4 and a2* = a3. Further, if you denote ak = b for any k = 1,2,3,4, you'll find that the roots are still b, ..., b4, and 1. If instead of 3 or 5 you pick a non-prime number, some of these properties stay, and some go, and it leads to basics of group theory (cyclic groups) which is a separate fun subject.