hi just visiting not an aryan intellectual
a is 0, 2pi/5, 4pi/5, 6pi/5, 8pi/5
cos theta + i sin theta is r=1, angle=theta in polar coordinates: the point on the unit circle with angle theta (bc cos^2 t + sin^2 t = 1)
Complex multiplication adds the angles and multiplies the radii, so the fifth roots of 1 are 1/5 around the unit circle, 2/5 around the unit circle, etc. When you take the fifth power they end up 0 around the unit circle
Now I'm not sure how you get b algebraically (update post if you find out), but I did it kind of geometrically?
We know (z+1)/z needs to have radius 1 and angle 2kpi/5 because those are the solutions to the fifth root of 1. Because it has radius 1, z+1 needs to have the same radius as z. Since their imaginary parts (y coordinates) are the same, they need to have opposite real parts: Re(z + 1) = - Re(z). We're flipping z around the y axis. Thus Re(z)=-1/2 and Re(z+1)=1/2.
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