WEBER'S CLASS NUMBER PROBLEM IN THE CYCLOTOMIC ℤ₂-EXTENSION OF ℚ, III

Abstract

Let h n denote the class number of [Formula: see text] which is a cyclic extension of degree 2 n over the rational number field ℚ. There are no known examples of h n > 1. We prove that a prime number ℓ does not divide h n for all n ≥ 1 if ℓ is less than 10 9 or ℓ satisfies a congruence relation ℓ ≢ ± 1 (mod 32).