5-Class towers of cyclic quartic fields arising from quintic reflection

Abstract

Let ζ 5 be a primitive fifth root of unity and d = 1 be a quadratic fundamental discriminant not divisible by 5. For the 5-dual cyclic quartic field M = Q((ζ 5 − ζ −1 5 ) √ d) of the quadratic fields k 1 = Q( √ d) and k 2 = Q(