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Congruences of Ankeny-Artin-Chowla type for pure quartic and sectic fields

Ankeny, Artin and Chowla [1] showed that there are congruences between class numbers of real quadratic fields and generalized Bernoulli numbers. Recently, Ito [3] has extended their results to the case of pure cubic fields using generalized Hurwitz numbers of Lichtenbaum [4]. In his paper, he suggested that similar results would be obtained for pure quartic and sectic fields. In this paper, we carry out this by following his idea. To give a congruence in an exact form, we need an idea due to Matthews [5]. As the argument in the sectic case is quite parallel to that in the quartic case, we shall discuss the former case briefly in the last two sections.