Note on class number parity of an abelian field of prime conductor

Abstract

Let n ≥ 1 be an integer and let 2 be the highest power of 2 dividing n. For a prime number p = 2n` + 1 with an odd prime number `, let N be the imaginary abelian field of conductor p and degree 2` over Q. We show that for n ≤ 30, the relative class number hN of N is odd when 2 is a primitive root modulo ` except for the case where (n, `) = (27, 3) and p = 163 with the help of computer.