Indivisibility of the class number of a real abelian field of prime conductor

Abstract

For a fixed integer n ≥ 1, let p = 2 nℓ + 1 be a prime number with an odd prime number ℓ and let F = F p,ℓ be the real abelian field of conductor p and degree ℓ . When n ≤ 21, we show that a prime number r does not divide the class number h F of F whenever r is a primitive root modulo ℓ with the help of computer. This generalizes a result of Jakubec and Mets¨ankyl¨a for the case n = 1.