On the Class Groups of Certain Real Cyclic Fields of 2-power Degree

Let e ≥ 2 be a fixed integer, and let p = 2e+1q + 1 be an odd prime number with 2 ∤ q. For 0 ≤ n ≤ e, let kn be the subfield of the pth cyclotomic field Q(ζp) of degree 2n. For L0 = Q(√2) or Q(√2ℓ) with an odd prime number ℓ, we put Ln = L0kn. For each 0 ≤ n ≤ e − 1, we denote by Fn the quadratic subextension of the (2, 2)-extension Ln+1/kn with Fn ≠ Ln, kn+1. It is a real cyclic field of degree 2n+1. We study the Galois module structure of the 2-parts of the narrow and the ordinary class groups of Fn. This generalizes a classical result of Rédei and Reichardt for the
case n = 0.