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

Abstract

Let p be a prime number of the form p = 2 ℓ +1 with some odd prime number ℓ . For such a prime number p , it is shown that the relative class number h (cid:0) p of the p th cyclotomic (cid:12)eld Q ( (cid:16) p ) is odd when 2 remains prime in Q ( (cid:16) ℓ ) + by Estes [3], Stevenhagen [11] and Mets(cid:127)ankyl(cid:127)a [8] using a Bernoulli number associated to Q ( (cid:16) p ). In this note, we give an alternative proof of the assertion using a cyclotomic unit of Q ( (cid:16) p ) + .