On class numbers inside the real pth cyclotomic field

We fix an integer n ≥ 1, a prime number with 2n and an integer s ≥ 0. We deal with a prime number p of the form p=2nᶠ+1. For 0 ≤ t ≤ f, let Kₜ be the real cyclic field of degree ᵗ contained in the pth cyclotomic field, and let hₜ be the class number of Kₜ. We show that when p (or f) is large enough with respect to n, and s, a prime number r does not divide the ratio hf/hf-(s+1) whenever r is a primitive root modulo ².