Infinite Class Field Tower of Some Non-abelian Number Fields

We generalize Schoof’s theorem in 1986 and apply it to construct a class of Kummer extensions of cyclotomic fields with an infinite class tower. Moreover, we construct some "small-size" non-abelian number fields with an infinite p-class field tower for p = 3, 5, 7. Here, "small size" is regarded as either a smaller root discriminant, fewer ramified rational primes, or both.