Field discriminants of cyclotomic period equations

We show that several orbital equations and orbital clusters of the quadratic (logistic) map coincide surprisingly with cyclotomic period equations, polynomials whose roots are Gaussian periods. An analytical expression for the field discriminant of period equations is obtained and applied to discover and to fill gaps in number field databases constructed by numerical search processes. Such expression allows easy assess to inessential divisors of conventional discriminants and sheds light into why numerical construction of databases is a hard problem. It also provides significant information about the organization of periodic orbits of the quadratic map.