On the class groups of certain imaginary cyclic fields of 2-power degree

Abstract

Let 𝑝 be an odd prime number and let 2^𝑒+1 be the highest power of 2 dividing 𝑝 − 1. For 0 ≤ 𝑛 ≤ 𝑒, let 𝑘_𝑛 be the real cyclic field of conductor 𝑝 and degree 2^𝑛. For a certain imaginary quadratic field 𝐿₀, we put 𝐿_𝑛 = 𝐿₀ 𝑘_𝑛. For 0 ≤ 𝑛 ≤ 𝑒 − 1, let ℱ_𝑛 be the imaginary quadratic subextension of the imaginary (2, 2)-extension 𝐿_𝑛+1/𝑘_𝑛 with ℱ_𝑛 ≠ 𝐿_𝑛. We study the Galois module structure of the 2-part of the ideal class group of the imaginary cyclic field ℱ_𝑛. This generalizes a classical result of Rédei and Reichardt for the case 𝑛 = 0.