I have a long-standing interest in determining all power bases for cyclotomic integer rings. Generators of power bases for arbitrary number fields appear to be randomly distributed, but for cyclotomic fields they exhibit a nice structure -- up to integer translation, the generators lie on the unit circle and the line Re(z) = 1/2 in the complex plane (the picture that you just clicked on).

References:

L. Robertson, Power bases for cyclotomic integer rings, J. Number Theory 69 (1998), 98--118.

I. Gaal and L. Robertson, Power bases for prime-power cyclotomic fields, J. Number Theory 120 (2006), 372--384.

G. Ranieri, Générateurs de l’anneau des entiers d’une extension cyclotomiqe, J. Number Theory, to appear.

 
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