On class numbers inside the real pth cyclotomic field

抄録

We fix an integer n ≥ 1, a prime number ℓ with ℓ 2n and an integer s ≥ 0. We deal with a prime number p of the form p = 2nℓ^f + 1. For 0 ≤ t ≤ f, let K_t be the real cyclic field of degree ℓ^t contained in the pth cyclotomic field, and let h_t be the class number of K_t. We show that when p (or f) is large enough with respect to n, ℓ and s, a prime number r does not divide the ratio h_f/h_{f - (s + 1)} whenever r is a primitive root modulo ℓ².