On Divisibility of the Class Number of a Real Abelian Field and Fermat Quotients

Let r be an odd prime number, and let K be a real abelian field of conductor f K and class number h K .When r f K , Jakubec (Abh.Math.Semin.Univ.Hambg.63 (1993), 67-86) and Metsnkyl (Manuscripta Math.93 (1997), 481-498) gave a sufficient condition for r h K in terms related to Fermat quotients associated to cyclotomic elements m -1 for integers m dividing f K .We give a refined version of this result by different methods.