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Cyclotomic and Gauss Periods

caledonia
in Analysis and Calculus

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Gauss periods, used to solve the heptadecagon, are defined in such a way that each level can be determined from the one above.  For example Gauss periods of length 4 determined from those of length 8.

Can anyone suggest how Gauss came to this conclusion.  Or offer a proof ?

Thanks

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