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

Let p be a prime number of the form p = 2ℓ+1 with some odd prime number ℓ. For such a prime number p, it is shown that the relative class number hp- of the pth cyclotomic field Q(ζp) is odd when 2 remains prime in Q(ζℓ)+ by Estes [3], Stevenhagen [11] and Metsänkylä [8] using a Bernoulli number associated to Q(ζp). In this note, we give an alternative proof of the assertion using a cyclotomic unit of Q(ζp)+.